I found my summer driving hat at an "antique mall" that had a large number of vendors in one location. There were a number of very suitable hats, but I had a challenge finding one to fit my big head and thick hair...I considered slitting the back and inserting elastic, but then I found one that fit.
Remember that you don't want a wide, floppy brim, so no Kentucky Derby hats! On the other hand, a bare bones hat can be dressed up or tied into your turnout with tasteful ribbon or perhaps an artificial flower (my SIL, the floral designer, would kill me for even saying artificial...)