Since you don't have a location in your profile it's hard to give clear advice based on geography, terrain and quarry. If you told me that you were in Northern BC my recommendation would be different than if you were in Southern Saskatchewan or the Maritimes. Are you a moose hunter or primarily deer? Do you play in the bush where grizzlies may be lurking? Do you need a little more oomph than your 7mm-08 or a lot more hitting power? If you only need to step up a little and you're recoil-shy then you should be looking at a 30-06. In reality a 338-06 is not a "big" caliber, nor is the 35 Whelen or the 9.3x62 (366 Wagner for the uninitiated). In reality these rifles can all be shot well and their recoil, while a substantial increase over the short-action chamberings, is not significant to a seasoned rifleman. Too, you aren't a handloader and buying "a case or two" of factory 338-06 hoping that it will shoot in your rifle is a big costly gamble.
As sad as it is, you need a 30-06 more than you need what is for all intents and purposes a wildcat. True it was legitimized to an extent, but it's only a miniscule amount more alive than the 6mm Lee Navy. Unless you handload, stay mainstream.