Chicago to Poland - Cheap

Looking to travel to Poland cheap in June - June 1-June 16 to be exact. LOT is obvious best choice for direct flight at $1,370 (even compared to those with a stop over), but I'm hoping to save a couple hundred bucks by getting a good deal to another European airport/Country and then continuing my travels with the likes of RyanAir or German Wings. Question is, where to fly to to get the best deal? Does anyone have recommendations on websites or airlines?

Thanks in advance for any advice.

June is the single most expensive month to travel - anywhere.

Air Berlin to Warsaw via Berlin on June 2nd for $1032.

For instance Chicago to Dublin with Aer Lingus (from ~$620 booked now) then Dublin to various cities in Poland (Krakow, Warsaw, Wroclaw, Gdansk etc.) with Ryanair (under $120).