Most Alaskan bear guides will want you to carry at least a .338 Magnum, and that can be a good idea. A 9.5-foot brownie I shot on a beach at Admiralty Island fell at the shot, then got up and ran into the brush - after sunset - and it was pretty hairy following him into the dark woods. I was glad I was carrying an M98 in .45-70 with heavy loads. We found him dead 25 yards in, but I would never have followed a wounded bear with just a .30-06. On the same hunt a fellow hunter hit his bear with five, 220-grain loads out of his '06 and the bear trotted off, only falling when hit again by several .340 Weatherby bullets.
Lots of "experts" who have never hunted anything bigger than a skinny whitetail will tell you that you can kill a brownie fine with a .30-06, and that is true. What they fail to realize is that if you have to follow a wounded animal into thich brush, the .30-06 feels like a .22RF. As a contingency cartridge for a wounded brownie, it stinks.