This isn't a totally comprehensive answer, but I can do a bit better than AD 250. In roughly 390 BCE, Thucydides writes his History of The Peloponnesian War, a classic quasi-historical account of the war between Athens and Sparta that happened in 431 BCE. Now the precise historical accuracy is not something that we need to discuss (nor is the question of if it should be read as a history, probably not.) However, at the beginning of the book, in his "Archeology" he outlines the history of the Athenian and Greek people, and he starts with a very early, non-settled population.

A few quotes (from "The Landmark Thucydides" trans. Strassler):

"For instance, it is evident that the country now called Hellas had in ancient times no settled population; on the contrary , migrations were of frequent occurrence, the several tribes readily abandoning their homes..."

"Without Commerce, without freedom of communication either by land or sea, cultivating nor more of their territory than the necessities of life required."

"Before the Trojan War there is no indication of any common action in Hellas, nor indeed of the univeral prevalence of the name."

(Hellas is the greek name for the peninsula, and the Ancient Greeks called themselves Helens)

Thucydides then goes on to give an account of how the city-states developed their political power and a number of other things.

From this we can see that Thucydides recognized that the Ancient Greeks were not a perpetual people, living forever in the same place (as most ancient civilizations seem to have an origin story that involves arriving and settling a location, but I don't claim to be an expert on that). He didn't have a full concept of hunter-gathers as we do today, but is thinking more of quasi-nomadic farmers that travel from place to place seeking safety.

I can't speak to whether they had any idea of a forager->neolithic farmer->bronze age->iron age progression, but they might have been aware of hunter-gatherers. Tacitus describes "Fenni" who live in Northern Europe, use bone tools, and hunt and gather for food. A popular idea is that he was describing Sami peoples, indigenous Scandinavians who at the time lived as hunter-gatherers and fishers (in medieval times they adopted reindeer pastoralism). The Sami are ethnically related to Finns and the word "Finn" was originally a Germanic exonym for the Sami, not the Finns- one proposed etymology is that it comes from the same root as find, indicating that "Finns" found (foraged for) their food rather than raising it themselves.

Unfortunately, given Tacitus's fanciful descriptions of other cultures it's hard to know whether his description was actually a garbled description of Sami or whether it just coincidentally lined up with reality. I'm not a classicist so I don't have the necessary know-how to navigate that debate. Plus, on the Scandinavian side, I've always been disappointed with how little info I've been able to find on first millennium Sami and their interactions with Germanic peoples (if anyone knows some good sources...?).