I have a lot of rifles, in a whole lot of different calibers, and I LOVE to use them all. Lots are smaller than .308, and many are bigger than .308. And that may be the main reason that of all 10 or so 30-06's I've owned, I have only ever hunted one of them, and only killed one critter with it, a big New Mexico bison that fell to a 180 Partition. I've just been too busy playing around with various Gucci calibers to bother with the old '06 in the hunting fields. Because I already KNEW how that was going to turn out. Though I'm far from the most experienced hunter, I have killed dozens of critters, including mule and Coues deer, pronghorn, elk, and the one bison (moose, sheep, and goats are still on the bucket list). So, in my probably about average experience level, the answer to the OPs question is:
Nothing.
I turn 60 next week and am sitting around looking at a pile of rifles, picked up at great deals over the years, that I may never get to hunt with. Two of my 30-06s just got sent to JES to become something bigger and another is going to be rebarreled to .256 Newton. So I'm gonna have even more "Gucci" rifles to hunt with, in my remaining years. I can't say that the 06s are going to get a whole lot more love, but there's not a better cartridge I "need."

All the best,
Rex