SkyTeam and Star Alliance offer San Francisco to Athens for around $675 to $700 for departures in February and March. SkyTeam Alliance partners Air France and KLM are the primary carriers for winter departures at $685 to $699. Star Alliance is primarily Swiss and Lufthansa and fares are as low as $675 round trip.
Then, April 2016, Delta drops fares to $645 round trip for one-stop flights connecting through New York JFK. There are about 7 departure dates in April 2016 with this $645 fare and another ten or so departure dates at $667 to $677 with Delta and Air France/KLM. Fares rise to around $1,400 round trip for May departures.
One potential option with these fares is to book a round trip departing in April with a July return date, then use miles to book flights home in April and back for summer travel. The most difficult part of summer award travel to Europe is finding an award seat back home. This strategy can get you to Greece two times with an April trip and a summer trip with one paid low fare ticket and one miles award ticket.
Delta Air Lines San Francisco – New York JFK – Athens ATH
$644.96 Tuesday April 5 – Tuesday April 12