Besides of course Homer's references that can be found in later works and commentaries, there's one interesting Solon's law, possibly of early 6th c. BCE, but survived by Diogenis Laertius [1.57].

According to it, the reciter of Homer's epic poems, who was substituting the previous one, should continue the recitation from the point where the last one stopped. Probably to avoid repetitions, as the poems were long, while the law was setting [or considering as precondition] the substitution of the reciters, that would keep the quality of the recitations high [*there's some debate on the translation of a term]. But I think that it's also indicating that Homer's epics had a central part in Athens's entertainment and they should be considered frequent, as a law was needed to regulate these.

There's also, on the opposite aspect, a story told by Herodotus [5.67], where Cleisthenes the tyrant of Sicyon of the early 6th c. BCE forbade the recitation contests of the Homer's epics, cause they were praising the city of Argos, against which Cleisthenes was fighting at the time. Again I think that it's signifying some importance, mentioning also the contests; for the age of Herodotus, too, as the story was reproduced then.

Of course both incidents took place before the classical era, but Solon's laws and the contests, I think, survived for some time.