Doc: I have hunted whitetails with my .30-06 for 29 years and I have to agree with the recommendation Dr. Rob gave you.
Most years I kill at least 3 deer; sometimes 4 (the limit) sometimes 2.
I have tested Winchester, Hornady, S&B, Remington and Federal in various bullets and weights.
In my gun the best grouping cartridge is
Federal Premium 165 grain Sierra GameKing boat-tail bullets.
The .30-06 is a great hunting round. I have never failed to recover a deer shot with this gun. I did shoot some of the deer more than once--but better take'm down while you can than to not recover your deer and wonder where that first shot hit for the rest of your life.
This is especially true of ELK. Nine times out of ten, that 2nd shot will be unnecessary; but ELK are tough and can run a loooooooooong way, even when fatally hit---so if he is still standing---shoot him again. You will never be sorry to find an elk with 2 holes in him. You might be very sorry to not find an elk with one hole in him.
Last year the 165 grain GameKing went right through an 8-point and demolished his heart and lungs. Broke a doe's neck--completely--severed the whole spinal column and actually disconnected and rearranged it like she had been hit with a running chainsaw.
I would have every confidence that the GameKing is sturdy enough for elk. But you will need to test it in your gun to see how it shoots from a first-shot cold barrel and how it shoots in a fouled barrel.
I am fortunate that my point of impact for a cold barrel is almost the same as the fouled barrel. Rifles do vary in this regard. Shoot and see.