Good ol' 30-06 with a good 165 - 180 gr bullet just plain works.
My standard load for the 30-06 is a 165 gr Nosler Ballistic Tip over a max charge of H4350 for over 2900 fps and good accuracy.
Had one heck of a season in 2016, and used that combo for:
cow elk @ 338 yards, one shot
black bear @ 325 yards, one shot
mule deer @ 140 yards, one shot drop, but needed a finisher when I got to the buck
antelope @ 245 yards, one shot
I've used the same load before and after. It's just a dandy. The 165 Ballistic Tip worked great on the cow elk (through one shoulder blade, destroyed the lungs, hit and broke the other shoulder blade, but never recovered the bullet). But... I think I'd prefer a 165 Partition, or a heavier partition for a dedicated elk load.
I went with 200 gr Partitions over H4350 for 2600 fps for my grizzly hunt. That was a real nice shooting load! Accurate and the recoil was just a nice soft push. I liked it so much that I considered doing all my hunting with that load - but I have a small mountain of 165's on hand, so I'll likely save the 200's for heavier game. Oh, the 200 not only did well on the grizzly, but also opened up nicely on a wolf at about 200 - 250 yards.
Years ago I used the 180 gr Nosler Partition - and it too worked well, before I dropped down to the 165's.
Shooters Pro Shop is a great source for Nosler Partitions, Accubonds, Ballistic Tips & more.
I'd recommend developing a load you like, stock up on those components, practice a lot with it before your elk hunt. Being in good shape, and being able to accurately place your shot, quickly, are both way more important than which particular bullet & powder charge you select.
Regards, Guy