If you need a soft recoiling gun, and the intended game is whitetail deer sized game and smaller, look no further than a .243 Winchester. The only people that have trouble killing whitetail deer with a .243 are writers for gun magazines!
If you have to add Elk into the mix, then I would look no further than a 7mm-08. However, the .260 Remington, the 6.5x55mm Swedish Mauser, and the 6.5 mm Creedmoor would also be contenders. If you can hunt elk in your state with a semi-auto, then the 6.5 creedmoor becomes a real player. It's available in AR platforms, and the semi-auto operation should help reduce felt recoil.
If you don't hand-load I would definitely go the .243 route, or 7mm-08 route.