Yes... the 338 would be good for loooong range hunting/sniping. Downside... heavier recoil, quite a bit more expensive, very difficult to find factory ammo (in out of the way places)
30-06 is a classic, do-everything round. Great for everything but perhaps the big bears, and it will work on them, if necessary. Downside... long action versus short action.. about 3/4" difference, max.
308... ballistic twin to the '06, can be chambered in a short action rifle, has been regularly used in 1000yd target matches, as has the 30-06. Very likely the easiest to find ammo for, but probably tied by the '06. Nearly everyplace sells 30-06, 270, 243, 308 ammo. Cheapest to shoot, especially if using surplus ammo.