If someone handed me a 300 Win Mag, the first thing I would want to do is reduce the recoil. I'm a big guy. I'm not recoil-adverse, but I also feel like I don't need to take a beating if I don't need to. With whitetails, I don't need to. All my rifle loads are downloaded to one degree or another.
Here's the data from the Hodgdon site for a minimum load of 165 grain bullet using H4895.
165 GR. SPR SP Hodgdon H4895 .308" 3.340" 55.0 2839 46,700 CUP
2839 FPS at the muzzle would be faster than either my 308 WIN load or my 30-06 load, but the pressure would be less. In other words, you could conceivably tame this beast down to where it is still shooting faster than a 30-06 and generating only slightly more recoil. For whitetails inside 200 yards, this would be plenty, and you would not have to use premium bullets either. At this velocity range, I'm using Hornady SP Interlock and Remmie Corelokts.
I would suggest H4895, because it is has two qualities that work well in this situation. First, it has a fairly gentle recoil owing to the way it burns. Second it produces a fairly even velocity no matter how much spare room is in the case or where it is situated in the case when the primer goes off. Hodgdon uses it in all its youth load data.
Remember that elk are not bullet-proof. They are just . . . er . . .obstinate. The forces and the damage required to kill them is not all that much more than a whitetail, but elk will continue to function longer before caving in. People tend to forget that. It's also the range. Whitetails are generally encountered inside 200 yards. Elk may require a much longer shot.
One other note: If you're just getting into reloading, downloading your hunting loads is a much safer and cheaper. I am on a lot of sites, and I see guys trying to get 30-06 performance out of there 30-30, and I cringe. Staying 5% or more off the MAX load does a bunch of things: it lowers per-round costs, it saves wear and tear on the rifle. It also reduces the chance of something going wrong. The closer you are to MAX, the easier it is to load a pipe bomb if you screw up. Whitetail deer certainly won't notice the difference.