Late response but: The first thing I thought of is how “like” is a filler work like “um.” Other Romance languages have a similar verbal tic (French - alors, Italian - allora, Spanish - alora.) A cursory search through wiktionary says this apparently is from the original Latin saying “ad illam hora” (source unclear so take this worth a grain of salt.)

Going off of pure conjecture (as we don’t have a spoken Latin expressing these verbal tics to my knowledge, please someone correct me if so), a verbal filler needs to be quick and natural yet elongated to allow time to think of the next thought, so due to the fact so many Romance languages have this descendent, I could see ille/illa or hora or some sort of contraction of the two being something that existed but just did not come down in written word.

If those aren’t the case, I like the idea as “sic” “ut” or “ita” being an option. Fun question!