I guess everyone knows Plato's allegory of the cave. He assumes people are in caves and then, he suggests that there is a possibility of "going out of the caves", gaining several nice properties with this. Do we have evidence that he noticed that we could "get out of a cave" and still be inside another "cave"? And that there could be an arbitrary number of nested caves?
That is your proposition and not Plato's. But I have to tell you the rule of the cosmos, that is "There is beginning, there is intermediate, and there is the end". One has begun to achieve the end trough intermediates. In fact, there is no continuity just as you want it to be in an infinite process, but this process of getting out of the darkness and reaching the light happens in everything and in every moment, therefore you never get out of the cave to enter another cave, although you may have come out of the cave a thousands of times.
If Plato's metaphor was inspired by the contrast between dreaming and awakening (not too big of an "if," but still), then no, he didn't notice that his metaphor could've naturally incorporated layers of caves modulo the notion of "false awakenings" in dream theory.
Then again, I don't know what the "state of the art" was in dream theory back then, AFAIK mystical interpretations of dream content and even the dreaming state itself were fairly common at the time. How much analysts tracked concepts like lucid dreaming, then, I don't know (lucid dreaming as a phenomenon seems to undermine supernaturalistic theories of the dream state, since being able to "reprogram" dream contents at will lends itself to viewing those contents as primarily subjective).
For what it's worth, weak sleepers-awakening analogies in intellectual self-development remain a blight on our society, at least insofar as "conspiracy theorists" use these analogies to justify their apophenia as equivalent to the "awake" state. But Plato would not necessarily have been very sympathetic to conspiracy-theorism by the by (he might've shared some premises, like distrust of government on account of what the Athenian regime did to Socrates, but he also seems to have favored a kind of elitism that is significantly, albeit not exclusively, at odds with how a lot of conspiracy theorists portray themselves). Moreover, then, however, had the Western history of ideas operated on footnotes to a Plato who recognized the peril of layered false awakenings, perhaps apophenics and their mental kindred would be less quick to believe that learning "the truth" about one layer of social dishonesty meant catapulting themselves directly to "the truth" in itself; perhaps instead, that is, they would've been more cautious about the next layer they found themselves on.
But I'm not Karl Popper, I don't want to blame Plato that stridently for the state-of-affairs of our world right now.