The elk that my family and I have taken so far number 77, just under 4 per year since 1994.
The majority of our elk were taken with .270/130 grain premium bullets, .308 and regular 180's, .30-06 and 165-180's, .35 Whelen and 250's, and the rest with a wide variety of other stuff.
Elk shot through the lungs with most any good bullet with sufficient weight/speed/construction run for about 40 yards and then die. They don't die any quicker from bigger, faster bullets in the lungs. Hit in the spine they drop like a rock, no matter what the rifle/bullet combination. Hit only on the edge of one lung, or in a non-vital area, and no matter what rifle and cartridge, they at a very high risk of being lost. They are far tougher to recover than moose would be with the same wounding shot.
It is difficult not to dwell on exceptions because they are so spectacular. I have personally seen elk killed instantly with a broadside high lung shot 140 grain Rem core lockt from a 7mm-08, and have seen a 260 grain Nosler Partition / .375 H&H deflect outwards and fail to get to the vitals after it hit the shoulder joint (Humerus/scapula) on a quartering towards angle.
You want a "dedicated" elk rifle. Ardent makes a good point about the .300 Win mag. But it would be nearly impossible to tell a difference in performance between it and your 30-06 in actual field conditions.
I prefer to shoot right through an elk when I can. When things go a little wonky it is reassuring to have them leaking blood out both sides - at least with my tracking skills. Where I have found an advantage of one cartridge over another is when heavy bone is hit or when an acute angle means you need a lot of penetration. Then, ( and despite the exception I noted above with the .375) a heavy bullet at moderate speed really does help penetration and killing power. I have seen this time and again with my favourite .35 Whelen shooting 250 grain grand Slam bullets at 2500 fps. they had sufficient mass and momentum to break the humerus and continue on into the lungs and out the other side, important if you want to recover your hit elk and not just shoot it. Monometal ( copper) bullets also help penetration. Based on my experience, penetration and bone crushing power without deflection trump speed and extra-long range capabilities.
Fast magnums do not perform well with barrels of 22", and are at their best with 26". But those long rifles are a pain to handle in the bush, in a scabbard, or in a blind. Huge high power scopes are often no advantage either. A 3-9x40 is versatile, powerful enough and handy. I seldom use a bigger or more powerful scope than that, and often smaller and lighter is nice. The low power scope setting can be very important for quick close shots in the bush. I've shot a bunch of elk at less than 50 yards, some as close as 15. I was always able to stalk closer than 350 yards. Handling is a forgotten quality in too many rifles. Pick an elk rifle that handles well and that you can shoot quickly from all positions. If you can't do that, cartridge choice doesn't matter much.
If you can handle the recoil, and don't mind the size of the rifle, a .338 is an excellent choice. So is a .375. but they are a bit much for most shooters. So, I suggest that you consider a .35 Whelen, or better yet a 9.3x62, a cartridge that I have not personally hunted with but which seems just a tad better than the Whelen. They are a tiny bit more capable than cartridges shooting lighter, faster bullets. Load with a heavy standard bullet ( 250-285 gr) or a medium weight ( 225-250 ) copper bullet at 2400-2600 fps and go slay any elk that walks. And at any practical range, which is in reality a maximum of about 300 yards for most people.
