A lot depends on the rifle and action you prefer.
In a bolt action you have a much greater selection. Your tolerance to recoil, the weight of the rifle,
the range at which you might shoot and the type of game. Ammo availability may also be a factor.
There is a factory load out there that will do the job on any game from squirrels to elephants.

The .300 Savage is sweet to shoot in a bolt action and is fine for white tails.
The 9.3x57 can be loaded with 232 to 325gr bullets and is a very efficient bush gun.
The 8x64S when loaded to full potential can significantly better the .30-06 with the same amount of powder.
The 9.3x62 is a powerhouse with a great knock down punch for moose and elk with a little more powder than the '06.
The .404 Jeffrey can be made with a modified standard Mauser 98 action. Pedigree and performance.