Do you reload? If you don't then you'd probably be wasting time on a .260, 6.5 Creedmoor, or .22-250 for any kind of longer range work. .260 and 6.5 CM just aren't cartridges you'll find everyday unless your shopping online, and the .22-250 just doesn't use bullets heavier than 55 grains which aren't the best for longer range shooting. So if you don't reload I'd really look at the .308 or .243 as the best options for ammunition availability, but only because you don't want the .223. Although my daughter (10 years old) had been using the .223 and 55 grain TSX to hammer whitetail for the last two seasons.

I'm a die hard 6mm fan, I currently run a 6X47 aka 6mm-222RM for varmints, deer, and pronghorn, and a .243 Win for long range target shooting. My next one will probably be a 6XC when my .243 barrel craps out. My little 6X47 will push 70 grain Nolser BT's at 3100 fps out of a 20" barrel and only uses 26.5 grains of powder, which makes it a very efficient little killer of deer sized game and smaller. My .243 is a 1:7 twist that will run pretty exclusively all bullets 105 grains and heavier.