Great gun, but I'd prefer a .270. More so than that would be an AR .308. The bad thing about 30-06 & 270's are they're a good distance gun but shooting into brush will own that caliber. Your 35 remington is a good brush gun, but not great for distance. Ideally that's one of the reasons why the .308 has become more popular. It can shoot very nicely over distances and it's considered a brush round which means if it hits some brush it won't throw the path off nearly as much as a 270 or 30-06 would get thrown off.
Now granted I live in Louisiana and nowhere around here do you have more than say 100-150 yards of clearing, so my AR-15 wth my 2.5x scope can well take care of things for the opening if I choose to hunt with it, of course I'd use a .223 though. As a side note I have taken a deer out with a .270. I think my best kill was with the smallest caliber would of been when I killed a deer with a .17 HMR. I was hunting with a friend and I brought my 30-30 out but I dropped it when I slipped and busted the scope, so when we came up on the deer, only one gun was still able to shoot accurate and that's what I used.