We go to Greece every Oct. and for the most part the weather has been very good, sun, blue skies, mild temperatures, a little rain, some thunderstorms at night but never any washouts. We all know weather can be unpredictable but if you look at the weather averages for travel to Greece you'll find it to be a nice time to be there.
How many days to you have to travel outside of Athens? There are numerous options, mainland or islands.
You can rent a car and go down to Nafplio on the Argolis Peninsula and use it as a base to explore that part of the Peloponnese which is rich in archeological and historic sites. Nafplio is one of Greece's loveliest towns.
You could also drive down to the Mani Peninsula and base yourself in Kardymili which is a beautiful town, on the water, lots of charm and a great place to start your exploration of the Mani.
Or . . . go to an island or islands. Too many to mention but October will be perfect for finding lower costs, less tourists and still nice weather. We've swam in the Aegean in October and the water was warm, hardly a sole on the beach or in the case of Sifnos shared the beach with goats and an overfriendly cat!
Corfu as stated is way over on the west side of the mainland and your best option is to fly there. I've never been there so can't speak of it.
I have been to Hydra (day trip including Spetses) It's one of Greece's most beautiful waterfront with the town cascading down hillsides toward the port. It's more of a day-trip experience but there is nice walking along the coastline and if you walk uphill away from the waterfront it won't be as crowded. Again October will find most anywhere in Greece less touristed so it may just work out.
Check out Matt's Greek Island Guide for more info: