If you arrive on a domestic DL flight, you'll be airside in T4 at JFK. Can you then use the DL lounge (as an arriving passenger) or perhaps Centurion (if you have th right AmEx card, although they might be strict about the time you can enter and see you as a departing passenger. [There could also be issues if you ahve checked bags that must be rechecked due to different tickets/PNRs.]

Another alternative if you;'re stuck land side for a couple hours is to go to the TWA hotel (IIRC T5 stop on the little train that goes around the terminals) for a drink , snack, or just to visit the historical exhibits of crew uniforms (one level up from the lobby, on the same level as the full service F&B outlets and the gift shop, and also the historic offices/rooms (as you go toward the towers with the hotel rooms). There's a pool roof deck with a ridiculous cover charge and a historic aircraft outside that's been transformed into a cocktail bar, probably only after about 5 pm local time.