Burgas, Bulgaria (also on the Black Sea coast) should be next for an U.S. airbase. It's even closer to the Bosporous (about 125 miles), which is the chokepoint allowing access to the Black Sea from the Sea of Marmara.
Just to clarify, Sevastopol is now part of Ukraine, not Russia, but the Russians still have their Black Sea Fleet naval base there. It's a very prickly situation.