DL award levels vary from market to market from/to US these days (just as revenue fares do). If you reposition to JFK, D1 awards are available from 196K roundtrip (potentially less with Amex 15% discount if you can find DL metal in both directions with DL OZ award inventory availability, but that doesn't look possible in the summer). These award levels require mid-week travel. So if you want to travel around weekends, there will only be higher award levels available for booking.

The key is to find where there are cheaper D1 revenue fares as award levels tend to correlate with them. You can use Google Flights map view mode to check D1 pricing out of various US gateway cities to EU markets (there are currently $2.5K roundtrip D1 revenue fares on JFK-FCO and a number of other EU destinations).