With the assibilation *ti > *ci > si being one of the best-known innovations in Finnic, one would think it would have been researched to exhaustion long since. But there still seem to be new discoveries available.
The best-known examples of assibilation are paradigmatic alternations in inflection, either in nominals (e.g. Fi. kaksi : stem kahte- ‘2’) or verbs (tietä- : imperfect stem tiesi- ‘to know’); and instances affecting the overall shape of a word root (sinä ‘2PS’ < *tinä, silta ‘bridge’ < *tilta, asia ‘thing’ < *atja) or a suffix (kala-si ‘your fish’ < *kala-ti). However, cases in word derivation such that a morpheme boundary originally occurred between *-t- and *-i- seem to have been left with less attention.
One morphological category where we could suspect previously understudied examples of assibilation hanging around are iterative verbs in -i-. That assibilation can take place in these is not news per se: at least one clear example has been known for long, namely sortaa ‘to break down, oppress’ → *sorta-j- > *sorti- > *sorci- > sorsia ‘to tease’. This appears to be the only example in modern Finnish where an underived and unassibilated verb stem still clearly survives alongside an assibilated one, though.
A bit more common are examples derived from nominal roots ending in -si : -te-. Here it is possible to however consider later derivation from the nominative singular or from the plural stem (uusi(-) + -i- → uusi-), instead of Proto-Finnic derivation from the oblique stem (*uutə-j- > *uuti- > *uuci- > uusi-). At least the first two verbs seems to have quite limited dialect distribution, and so are probably not independent examples of assibilation.
- kirsi ‘frost’ → kirsiä ‘to soften when thawing (of the ground)’
- korsi ‘culm’ → N. Krl. koršie ‘to grow longer (of grain)’
- kynsi ‘nail’ → kynsiä ‘to scratch’
- niisi ‘heddle’ → niisiä ‘to thread warps through the heddle’
- uusi ‘new’ → uusia ‘to renew’
At other times, assibilation is identifiable only by comparison with distant relatives or parallel derivatives. Three likely and one further possible example are found in modern Finnish (all involved etymological connections already appear in earlier literature, though they have not necessarily been explained through *-ti- > -si-):
- jyrsiä ‘to gnaw’: likely < *jürci- < *jürtä-j-, from unattested *jürtä-, in turn segmentable as a causative *jür-tä-. Known cognates elsewhere in Uralic (Permic *jɨrɨ-, Mansi *jär-; both likewise ‘to gnaw’) suggest that the basic root was simply *jürə-.
- kursia ‘to stitch together’: perhaps similarly < *kurci- < *kur-ta-j-, derived from the same root as kuroa ‘to stitch together, to stretch together’; perhaps an applicative derivative = *kur-o-. The basic root *kurə- has known cognates in Samic *korë-, Samoyedic *kur-å- (where *-å- must be a derivative element, per the mismatch with Samic and the absense of the regular sound change *u-a > *ə-å). 
- suosia ‘to favor’: likely < *sooci- < *soota-j- ← unattested *soo-ta- ← *soo- (> suo-) ‘to grant, to provide’.
- talsia ‘to walk slowly’: appears to be likely related to tallata ‘to tread’. However, assuming a common root *talta- has the problem that the latter verb shows unvarying -ll-, e.g. Veps tallata (not ˣtaldata). To uphold this connection, it would seem to be necessary to assume generalization of the weak grade -ll- somewhere in the western Finnic area, followed by diffusion of the newly reformed verb to the rest of the family. Also, we would actually expect *talta-j- > **taltoi-! Some kind of analogical formation therefore seems more likely than soundlawful Proto-Finnic development.
From Karelian I can additionally find viršie (Northern) ‘to dawdle’. If from *vir-tä-j-, this might be connectable with viruo (~ Fi. virua, etc.) ‘to lay about, be sick’.
A relatively similar scenario could be moreover crafted for Krl. polzie (Southern) ‘to crawl’, which seems in theory derivable from polvi ‘knee’; a Proto-Finnic intermediate derivative *polwə-ta- > *polw-ta- *polta- ‘to kneel’ would need to be posited. However, this is much more straightforwardly explainable as a loanword from Russian ползать ‘to crawl’…  and so what we gain here instead is a reminder about the unreliability of etymological connections built on multi-stage derivational assumptions.
A common thread in these examples however seems to emerge, which I think provides some extra backing for reconstructing unattested “intermediate” verb stems such as *jürtä-, *virtä- (your call if this is actually decisive). This is an avoidance of verbs of the shape **CVRi-, especially from base roots of the shape *CVRə-,  upheld by deriving the iterative instead from a causative or pseudo-causative extended stem, formed by the common verbal suffix *-tA-. I have no idea what motivation this constraint could have behind it, though.
I think there is also one other larger category of iteratives that show assibilation. These are verbs formed with a suffix -(e)ksi-, predominantly from basic intransitive verbs:
- haave ‘daydream’ → haaveksia ‘to daydream’
- imeä → imeksiä ‘to suck’
- istua → istuksia ‘to sit (around)’
- kantaa → kanneksia ‘to carry’
- kulkea ‘to go’ → kuljeksia ‘to walk about’
- kusta → kuseksia ‘to piss’
- lukea → lueksia ‘to read’
- niellä → nieleksiä ‘to swallow’
- nuolla → nuoleksia ‘to lick’
- olla ‘to be’ → oleksia ‘to stay at’
- pierrä → piereksiä ‘to fart’
- piillä → piileksiä ‘to hide’
- purra ‘to bite’ → pureksia ‘to chew’
- ripistä ‘(of rain or raindrops) to make noise’ → ripeksiä ‘to rain lightly, drizzle’
- seisoa ‘to stand’ → seisoksia ‘to stand around’
- surra ‘to mourn’ → sureksia ‘to be sad’
- sylkeä → syljeksiä ‘to spit’
- tunkea ‘to cram’ → tungeksia ‘to crowd, throng’
- töpätä ‘to make a small mistake, hit a snaggle’ → töpeksiä ‘to make a lousy job at smth’
- uni ‘dream’ → uneksia ‘to dream’
- vuolla → vuoleksia ‘to whittle’
Many of these seem to have developed a more durative than iterative meaning, but at least verbs like kuseksia, nieleksiä, pureksia, syljeksiä clearly refer to iterated actions. It’s also worth noting that again, none of these verbs have simpler -i-iteratives such as ˣimiä, ˣkusia, ˣnuolia, ˣsuria…
I also think that this group needs to be separated from a distinct group of “sensive” verbs, mostly derived from adjectives, indicating considering something similar to the base word. Unlike the above, these are transitive verbs coexisting with synonymous verbs ending in -(e)ksU-:
- halpa ‘cheap’ → halveksia ~ halveksua ‘to look down on smth’
- hylätä ‘to discard’ → hyljeksiä ~ hyljeksyä ‘to shun smth’
- kumma ‘odd’ → kummeksia ~ kummeksua ‘to wonder, be puzzled over smth’
- nyreä ‘grumpy’ → nyreksiä ~ nyreksyä ‘to be picky over smth, accept smth grudgingly’
- paha ‘bad’ → paheksia ~ paheksua ‘to disapprove of smth’
- vähä ‘few, small’ → väheksiä ~ väheksyä ‘to belittle smth’
Hakulinen in SKRK notes the difference as well, though drawing the separating line mainly on the basis of if the verbs in question are derived from verbs or from nominals (thus placing e.g. haaveksia and uneksia instead in the 2nd group). He suggests that the second group might be built on the transitive case, ending in -ksi (probably correct), while the first group might be built on the denominal suffix -s : -kse- seen in e.g. kutoa ‘to weave’ → kudos : kudokse- ‘weave, textile’.
What I find more promising is the possibility of deriving the first group’s compound suffix -ksi- instead from Proto-Finnic *-kci- < earlier *-kti- < *-ktA-j-, where *-ktA- is the preform of the common causative-transitive verb suffix -ttA-. In many cases we can indeed still locate such a derivative alongside -ksi-iteratives:
- imeksiä < ? *imektä-j- ~ imettää < ? *imektä- ‘to suckle’
- istuksia < ? *istukta-j- ~ istuttaa < ? *istukta- ‘to sit someone down; to plant’
- kanneksia < ? *kandëkta-j- ~ kannattaa < ? *kandakta- ‘to support, hold up’
- kuljeksia < ? *kulgëkta-j- ~ kuljettaa < ? *kulgëkta- ‘to transport’
- kuseksia < ? *kusëkta-j- ~ kusettaa < ? *kusëkta- ‘to feel like peeing, cause urination’
- lueksia < ? *lugëkta-j- ~ luettaa < ? *lugëkta- ‘to make someone read smth’
- oleksia < ? *olëkta-j- ~ olettaa < ? *olëkta- ‘to assume’
- piereksiä < ? *peerektä-j- ~ pierettää < ? *peerektä- ‘to feel like farting, cause flatulence’
- seisoksia < ? *saisokta-j- ~ seisottaa < ? *saisokta- ‘to make smth stand’
- sureksia < ? *surëkta-j- ~ surettaa < ? *surëkta- ‘to make/be sad’
- syljeksiä < ? *sülgektä-j- ~ syljettää < ? *sülgektä- ‘to feel like spitting, cause excess salivation’
- uneksia < ? *unëkta-j- ~ unettaa < ? *unëkta- ‘to make/be sleepy’
Since I am basically working here with the internal reconstruction of Finnish, rather than from properly comparative Finnic data, there is of course the risk that some of these verbs may have been derived secondarily, as simply root+ksi-. One particularly good candidate might be Fi. surra and its derivatives. These have taken on the meaning ‘to mourn, be sad’ secondarily from suru ~ surku ‘sadness’, which is a loan from Scandinavian (Old Norse sorg). The original meaning, preserved in e.g. Es. surema, is instead ‘to die’ — and we definitely do not expect a verb of this meaning to have had any original iterative (habitual, frequentative…) derivatives. Regardless, the existence of this general pattern at all seems like sufficient evidence to conclude that at least some examples here probably date to Proto-Finnic times already. I would bet in particular on the “secretion verb” group (kuseksia, piereksiä, syljeksiä) and the “consumption verb” group (imeksiä, nieleksiä, nuoleksia, pureksia), both of which are entirely built on common Uralic primary verb roots.
This etymology for the suffix -ksi- also has one interesting implication: it confirms that Finnic -ttA- indeed derives from earlier *-ktA- (as continued also in Samic *-ktē-, Mari *-kte-, Permic *-ektɨ-) and not from earlier *-ptA- (as continued in Khanty *-ptə-, Samoyedic *-ptA-). The representation in Mordvinic (*-ftə-), Hungarian (-t-) and perhaps Mansi (*-t-) remains ambiguous though, and hence it is unclear to me which form(s) of this suffix represent the original Proto-Uralic situation.
 Samic *koarō- ‘to sew’ also seems related somehow. If *kurə- were from earlier *korə-, we could consider the possibility that the sound change *oCə > *uCə was later than *-əw- > *-o- (*korə- : *korə-w- > *kurə- : *koro-), but the Samoyedic cognate with *-u- seems to render this impossible.
 As pointed out to me by Niklas Metsäranta.
 Note that iteratives or similar derivatives based on roots of the shape CVRA — e.g. Fi. kerä ‘ball of twine’ → keriä ‘to roll up’; pesä ‘nest’ → pesiä ‘to nest’ — would have still been diphthong stems such as *kerei-, *pesei- in Proto-Finnic.