I have a fair amount of experience with larger bores on the .30-06 case. These cartridges work by expansion ratios. It's the expanding gas which pushes the bullet, and as gas expands, the pressure drops.
In larger bores, the pressure will drop faster because of the increased volume. That, in turn, calls for more powder, which means more energy.
The first such large-bore wildcat based on the .30-06 case was the .35 Whelen. The Whelen's fault (if it has one) is that it is too much bore for the volume of the .30-06 case.
There are two solutions -- one was to reduce the bore size to .33, which was first done by O'Neill , Keith and Hopkins, using a British .333 caliber bullet to produce the .333 OKH. The modern .338-06 is virtually identical to this cartridge, but using the .338 bullet (which was not available when the .333 OKH was developed.) For elk, I like the 210 grain Nosler Partition Jacket in this rifle, driven to about 2,800 fps. That gives you about 20% more energy than a 180 grain bullet from a .30-06.
The second approach was by C. Norman Brown. He reamed out a .35 Whelen chamber, taking all the taper out of the cartridge and moving the shoulder forward and making it steeper, increasing case capacity about 11%. The .35 B-W (in my rifle) will drive a 225 grain bullet at about 2,800 fps.
The .338-06 is an easy cartridge to case form (if you can't get factory cases.) Simply load a .30-06 case with about 10 grains of Bullseye (you may have to experiment to get the charge right) and a quarter sheet of toilet paper (no bullet needed) in the case mouth. Chamber it, point in a safe direction, and pull the trigger. When you open the bolt, out will pop a .338-06 case.
Wear ear muffs.