I've only shot a few cows, so I'm no expert. I used a 30.06 (180) a .375 H&H (300 BT) and a .338 Win Mag ( 185 XLC). I loaded the Speer 160 HotCor for a friends .280 AI and he also shot a big cow with that load. all of them worked swell ( actually, the 300 BT made the biggest mess! I was going to use it in South Africa and was testing. I ended up using 270gr Winchester Fail Safes) I've shot about 25 head of Plains Game with mostly medium calibers, the 300WM being the smallest. I love those mediums on heavy game!
I agree that (a) Wyoming is "extremely" proud of their Non Resident tags, ha and (b) the elk could be anywhere from watching you take a poop to the next ridge over, ha. I like the 35 calibers for elk, based on much use on elk size animals in south Africa. I used my own beloved, treasured, never to be parted with Mod 700 35 Whelen AI with both the 250X and later the 200X. My closest shot was about 30 yds. on a Gemsbok ( 250X) and 347yds on a Black Wildebeest (200X) I just found a great load using the Nosler 250 PT so will use it next. Anything over 250yds is a long way! In elk country, its tempting to take a shot at elk further than one should. It doesn't have as much to do with the rifle as "finding where he was standing" when you made the shot. It isn't reasonable to count on him DRT, but he will most likely run a bit. In the prairie its not so bad, in the mountains, its a nightmare. Snow is a big help for blood, but not for tracks as a 100 elk could have walked there. Once you walk over to where you thought he was, everything looks different, etc. If you have a guide or a buddy, it helps to have him keep an eye on a landmark and watch as you walk over ( walkie talkie helps) to look for sign.
Now, with that in mind, your 35 Whelan AI is fine with the 250s ( about perfect in fact) the 200 TTSX is very flat shooting too. The 358 could work with the 250, but a 225 Partition or 200 TSX is more useful....know where both hit out to 300, and let that be your Max. There is an old saying that 90% of the elk spend 90% of their time in 90% of the thickest stuff! ha. I like to hunt the edges of dark timber/lighter timber and a light, fast handling rifle really helps. I use my 35 AI with a synthetic stock in Fall, (Leupold 1.5x5) and in its wood stock and (Bushnell 3200 2x7) in late hunts . I agree also with making double dog sure your scope is a good one! I still keep iron sights on my Whelan with QD rings, just because I had a high dollar Zeiss quit on me in South Africa on a lightweight .340 Weatherby! You guessed it, slick barrel. Thankfully my friend was using a 300 Winchester that I had given to him, so it was no stretch to finish out my hunt.
Oh yea, my first cow was shot running, 135 steps, through an Aspen edge, headed for the timber. The 30.06 180 spined her, popped her again when she was trying to get up. Second cow was right at 200yds, lounging with family at a beaver pond, open shot, with the 375. She humped up with a high lung shot. Since I was afraid she was going to run into a close, deep canyon, I popped her again. Could stick your fist in both exits; scratch the 300 sierra for that! My Third was on a late season hunt (think snow!) with the 185 XLC out of the 338WM at approx. 250yds, up on the ridgeline above me. Lung shot, she stumbled 10yds and dropped. I just slide her off the steep ridge toward the trail/ she hung up between two aspens and I thought I would have a heart attack getting her free! wow!
So yes, unless you are lucky enough to shoot them off a hayfield ( my personal preference!) you need to be in good shape and hire a guide or have some hunting buddies. Solo elk hunting is very hard on us old, fat guys, ha. Have a ball, but take care of yourself.
