Dave, if recoil is a problem, I'd stick with the .243! With good 100 grain bullets, the .243/6MM will take any whitetail you set your sights on!
Years ago I switched from the 6MM Rem. to the .308 and 7MM RM.
They are both great performers, but no deer that I have ever shot with either of these would have known the difference between them and the .243/6MM.
If you just want to move up in caliber, the .257 Roberts, .260 or 7MM-08 will give you great service, with little recoil! Good luck!