All three rounds will do the job you have planned with no problem in the hands of a competent shot at the distances where 99% of hunters would attempt a shot.

If you are prone to flinching or want to spend much time shooting the rifle after your elk hunt, the 300 would drop to the bottom of my list. The magnum ammo is markedly more expensive than the other two and the recoil becomes unpleasant for any sustained target shooting.

As far as choosing between the .308 or the 30-06, pick the one you like the best. The rounds are too close in performance to waste time debating about them.

I agree with Jerry above, your existing rifle will do the job just fine but there's certainly nothing wrong with shopping for another big caliber rifle if the funds permit.