Ed, I've got a 30.06 and it's now my elk and moose rifle here in B.C.. For this big game, after a lot of research and talking to a lot of guys with experience, I went with the 168 grain Barnes TTSX ammo. I've shot a few elk and several moose with it, and they've dropped immediately or didn't go far at all. It's a tough all copper bullet that stays in one piece even if you hit a large bone and penetrates deeply.
When I used the 30.06 for our big stout mule deer and smaller whitetails, I used a 150 grain Nosler Accubond in the later years. It's also a stout bullet but not as stout as the Barnes, and will open up perfectly out to several hundred yards. That makes it an excellent bullet for deer at close or long range, it shoots accurately out of almost every rifle, and the light weight reduces recoil. I'd recommend them loaded in Federal ammo.
Hope this helps! :-)