American and Oneworld partners have round-trip flights from Austin (AUS) to Athens (ATH) starting at $586 with a connection. Flights November to March. Holiday blackouts apply. 5-day minimum stay required. Expect prices to fluctuate by +/- $20. Flight may operate as a British Airways/Finnair code-share.
Note: Carry-on bag is allowed for free, but checked bags will cost about +$60 each-way. An upgrade into “Regular Economy” is only +4.50 more each-way at the time of this writing, you’ll probably want to do that instead of deal with the “Basic Economy” restrictions.