As a matter of fact the 30-06, the 7 mag, the 300 mag, and the 338 win mag are 2/10ths of a MOA apart at 500 yds. The 338 actually is good to go on big game at 800yds with a .515 BC bullet of 225 grains. Snipers in the nam shot as far as 1300 yds with an 06 the 338 win mag is capable of a full 2000 yds so long as you have the MOA up to do it. As far as the 308 being more accurate...... sorry guys I've shot 338 win's that group into .239 MOA so thats a wash too. MY advice is to develop a load w/ a 165 grain bullet or maybe a 185 grain. There is your varmint load/deer load. Then develop a 225 grain load with the premium hornady bullet to around 2800 fps. That'll take care of any bear that walks.
Semper Fi. Best of luck