Airforce Texan SS or LSS CF .457
I second this. I haven't been air gunning as long as a lot of these folks but have been tromping the woods since I could run away from mom.
45 or 50 cal are going to beat the 357, especially if you want just one for yotes to elk (yea, I know you said white tails), and if you are an elk hunter, you'll be scratching that itch sooner than later.
I would recommend a Texan in what ever heavy caliber you choose and the second gen air valve. If you get the carbon fiber wrapped tank, you are in business. I have the first gen and it is a beast. 3-4 full power shots is about it for a 457 cal whit 300ish grain slugs. Looking at 500-600 ft-lb muzzle energy.
I would use these for just about anything larger than a rabbit. 143 round balls can go to 10 shots at full power, and generate 250ish ft-lb. Fantastic coyote and even pig medicine. Cheap, accurate, and fun.
Lots of opinions, and that is mine.
Happy Shopping and welcome to the mayhem.