Short answer: Sure an '06 should be fine.
I've shot a few deer with the .30-06, and several more deer with a lesser caliber. For deer, the .30-06 has a lot of what Art Eatman calls "bang-flop". It's very satisfying to hit a deer, hear the hit, and see it go down like a light switch was thrown, and that (minus sometimes hearing the hit) is what has happened every time I've shot a deer with a .30-'06. But then again, I've had similar reactions to deer that I've shot with .257 Roberts loads. The .30-'06, then, is simply more than the minimum rifle that is needed (note: I did not say "too much gun").
So what is the '06 optimal for? Well, an elk is a (much) larger species of deer. Put an '06 bullet through an elk's lungs or heart or into its spine, and you get similar results. I submit that the .30-'06 is absolutely adequate for elk. When I called Sierra Tech Support asking about best loads for the 165g GameKing for elk, they assured me that .30-'06 velocities (2700-2800fps) were great for elk, but advised me not to use that bullet at .300 WinMag velocities (3000-3200fps), due to the possibility of over-expansion at the higher velocities. But Sierra absolutely endorsed their 165g bullet as an elk bullet out of the .30-'06.
"Welcome to The Firing Line, a virtual community dedicated to the discussion and advancement of responsible firearms ownership."T.F.L. Policy Page
Will you, too, be one who stands in the gap?