How about a .338 Lapua?
First I think you decide what class of cartridge you want, whether it's mid-powered (e.g. .308), magnum class (.e.g. 7mmMagnum) or super magnum (e.g. .416Barrett). Then decide on the type/style of rifle you want. And go seek that particular combination. Most any caliber can be had in any style, it's only a matter of what you're willing to pay.
All things considered, or at least some things, I decided, for anything greater than the .223, to stick with .308 case based rounds. .243, .260, 7mm-08, .308 and .338-08. A wide range of the mid-powered cartridges. The reason being the easy availability of the .308 brass for reloading. (Yes, I could have as easily chosen the .30-06; just wanted the shorter actions.) IF factory ammo availability is your ONLY consideration, then my observation is .30-06 and .270Win is the most common round on the shelves, after .223.
My recommendation in your situation is to go for the .308. It would be a good addition to your stable. You can go for an AR-10, but I'd recommend you get a solid, heavy barrel bolt action. Something along the heaver barreled Savage, Tikka or Remington line and top it with a quality scope.