I'd suggest looking into the following cals as well; 338 federal, 338-06, 35 whelen (already mentioned but worth mentioning again), and a 325 wsm These are all great cals and I own or have owned all of them with the exception of the 325 wsm. However, if you are looking for a significant increase in power from the -06 and are planning on shooting grizzlies, I'd go with the 338 winny. You load bullets as small as 180 grains and as large as 300 (plenty big for an inland grizzly). It's also flat enough for long shots.
A couple other cals to consider for the game you are pursuing would be the 375 H&H, 375 Ruger, 358 norma mag, the 9.3X62, and the 9.3X64. All heavier for cal bullets, but not as flat shooting as the 338 win mg. I haven't shot a 375 ruger, but it is in a standard length case (meaning lighter rifles for all that hiking), and I've heard great things about it on AR etc. I have a 375 H&H and a 9.3X74R, which is ballistically identical to the 9.3X62. Both are big enough for anything up to elephant, and neither have that much recoil in my opinion. The 9.3X64 is harder to get a hold of, in terms of rifles and bullets, but it's ballistics are impressive and similar to the 375 H&H. The same could be said of the 358 norma.
All that said, your best bet would be the 338 win mag in my mind.