The ancient Greeks did, in fact, wear different clothes in the winter versus in the summer. The Athenian historian Xenophon (lived c. 430 – c. 354 BCE) in his Memorabilia or Memoirs of Socrates 1.6.6 portrays Socrates as referencing this as a common practice while defending his own practice of asceticism in a conversation with Antiphon the Sophist. Socrates says, on page 96 of the Penguin edition, translated by Hugh Tredennick and revised by Robin Waterfield:

"As for cloaks, you know that people change them because of cold or hot weather, and they wear shoes to prevent things from hurting their feet and so impeding their movements. Well, have you ever known me to stay indoors more than anybody else on account of the cold, or compete with anyone for the shade on account of the heat, or fail to walk wherever I wanted because my feet were sore? Don't you know that those who are physically weakest by nature, if they train with a particular end in view, become better able to achieve that end, with less effort to themselves, than the strongest athletes who neglect their training?"

During the winter, many ancient Greek people probably wore the himation, a kind of long cloak that usually came down to the ankles and was made from a single large rectangular piece of heavy woolen fabric. The himation was worn by people of all ages, genders, and social classes. People sometimes wore it on its own without anything underneath it; other times people wore it over their usual chitons. You can see examples of statues from the island of Delos depicting a woman and a man both wearing himatia here.

It is likely that on particularly cold days during the winter, Greek people would huddle indoors around the hearth to keep warm. Vase paintings of Achilleus mourning for Patroklos sometimes show him completely wrapped in his himation from head to toe. (Here is one example, a tondo from a red-figure kylix dating to around 500 BCE.) We can imagine that some people might have wrapped themselves similarly during the winter if it was especially cold.

People also frequently wore multiple layers during the winter. If it was extremely cold outside, a person could wear multiple himatia and other garments over each other for extra warmth.

I wrote a blog post about how ancient people kept warm during the winter two years ago, which is fairly short, but covers the ancient Mediterranean world a little bit more broadly than what I've addressed here.

While you're waiting for a response, you may be interested in /u/toldinstone 's discussion of Roman tunics