First of all, I live in Alaska, as does Ratltrap, The Bear 78, and corbanzo. I've lived here for 37 years.
As others have said, depends on where you are going to be hunting. That depends mainly on where you are going to live, Alaska is a big state with seven differant geographic areas. Southeast, Rainforest. South Central, Temperate. Interior, Arid. The other two areas, don't ask!
The 30-06 you have is good, especially if you are going to be hunting in the interior. Around the Fairbanks area. At least until you have been here a while and decide yourself that you need more gun. If you are going to be living on Kodiak, I'd get something bigger. I personally carry a 30-06 a lot, especially during the winter. After the bears come out I also carry a .338. I usually carry two guns when I go out, I have been known to brake them, and once even lost one.
The most populer gun by far in Alaska is the .338 Win Mag. That's the gun that dumb ____ used to shoot a hole in the pipeline. And you can find shells for them all over the state. And it is adiquate for the BIG BEARS.
As Ratltrap suggest wood is not good here. In fact I have gotten rid of all my wood, on all my bolt guns. Reguler wood stocks will ruin a hunt fast. With the sudden change in humidity and temp. you are asking for trouble if you leave wood on any bolt gun. You may think you have it sealed, then after trudging around in rain for two days you finally shoot and it misses by three feet at 150 yards you want to kill yourself, let alone beat the gun to pieces.
Your 257 WBY will make a good Sheep and Goat gun. It will also be good for Caribou, if you don't run into a Grizzly.