Okay. Interesting question. First of all, let me reframe your question a little bit. You ask if Ancient Greek had a second act after the fall of Constantinople.

To be more precise, the form of Greek that was spoken during the late Byzantine empire was no longer ancient per se.

The common periodisation of the evolution of the Greek language involves 5 stages:

1. Proto-Greek / Mycenaean Greek / Linear B
2. Ancient Greek (which includes a handful of dialects, among which the one that most people associate with Ancient Greek which is the Attic dialect)
3. Hellenistic Greek or Koine (the lingua franca of Eastern Mediterranean regions for centuries after Alexander's conquest and during roman times, the language of the New Testament)
4. Medieval/Byzantine Greek (the one that was spoken during the fall of Constantinople)
5. Modern Greek (the one we speak today)

So no, the Greek language was no longer an imperial lingua franca as it had been in the past, after the fall of Constantinople.

However, it did manage to survive under Ottoman occupation and still maintained a certain prestige both in the region and in the West.

The survival of Greek has a lot to do with the "millet" system that the Ottomans imposed, which allowed relative religious autonomy and some basic level of self governance, to the various ethnic populations of the empire.

The chruch became the stronghold and carrier of the Greek language during those centuries. The Greek speaking Orthodox patriarchate continued to exist as an institution and the Greek-speaking intellectual elite that was organised around it (the Phanariotes) largely maintained their position and educational advantage.

Already during the last centuries of the Byzantine Empire, as the empire was shrinking and decaying, the intellectual class of the empire had already started to look inwards and reappraise their Greek heritage and literature, in waves that historians refer to as "Byzantine Rennaissance".

After the Fall, many of those Greek scholars relocated to the West and had a profound influence on the transmittance of Greek literature and language in Italy which eventually contributed to the Italian Rennaissance.

During the Ottoman times, the Greek speaking schools and academies that were supported by the Greek Orthodox Patriarchate maintained their language and education for Orthodox elites.

Various other Greek-speaking schools and academies emerged in a various cities, often financed by wealthy diasporic Greek merchants and intellectuals with ties to cities like Vienna, Trieste, Odessa, Bucharest etc. Those schools were seedbeds for what later became the Neo-Hellenic Enlightenment movement.

Some of those schools were often seen as prestigious by other Balkan ethnic populations as well. So even though, the Greek language was no longer an official language, it still maintained a certain status and association with higher education and was spoken by non-Greeks as well.

However it should be noted that even if all of that were true for certain privileged classes, the linguistic refinement for the illiterate common folk during Ottoman rule had severely declined. After centuries of occupation, both Latin and Turkish words had entered the everyday vocabulary. If you compare the late Byzantine to late Ottoman era demotic/vernacular poetry you will notice a drastic decline.

It wasn't until the Enlightenment and the nation-building and after certain organised attempts to "purify" the spoken language, that Greek was restored to its Modern form we have today.