If your hunting is 600 yards or less, the 308 will be fine. I personally would prefer a barrel at least 24" on the magnum, 26" optimum. You will trade off the velocity advantage over your 30-06 with the short tube. iMHO
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With the 22" I'd go with a 308 and great scope. After you become good at distances over 500 yards, then make a decision to go up with a rebarrel, you might jump right to a 338 edge or decide a 7 stw or a 7-300WSM for getting out into the 1200 range.
The 308 barrel may turn out to be a switch barrel if you don't shoot it out. Then you have 2 guns in 1.
Here's a thought. This will prevent you from having to cut your barrel back to far and give you a good cartridge for both target and hunting. The 300WSM, RCM, and SAUM all will work well in this application. I believe the SAUM and RCM are the better rounds for a shorter barrel. However all three will do what you want and more. The RCM and SAUM will use less powder than both the Win Mag, and WSM, but maintain the ballistic advantage of a magnum caliber. The reason I say you wont have to set the barrel back as far is due to all of these cartridges I have suggested, are the diameter of the rim on the Win Mag case. Take a look at the numbers. You could run a 168-185 for target. This will minimize the recoil. With the smaller case you could also do a reduced load for recoil management. Then when you are ready to go hunting, then stuff 185 Bergers or bigger (up to 215grn bullet) with a full power charge to slam your query of choice. Just going to put that out there.
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Great advice on the WSM, you are giving up to much over the 30-06 with the 308. Recoil on a 300 Winmag, I don't think so. With any reasonable weight rifle recoil will not be an issue. Breaks are nice to see the target but suck on noise and sometimes blast. You'll be fine.
Location: The rifle range, or archery range or behind the computer in Alaska
Re: Caliber decision
#1: You mentioned competition. Most competitions don't allow brakes.
#2: Felt recoil means different things to different shooters. To me, when I think of recoil in a hunting situation it is a non-factor. Sighting in, development and LR practice can be a real b!tch with a lot of recoil but in hunting you don't feel it. Is the 300WM that bad? Maybe. For me, itis too much for matches and the rest of the shooting I do. I shoot anywhere from 500-2000+ rounds a year. Believe me, recoil becomes a major concern. Most guys who say the 300WM does not kick that hard probably don't shoot matches or 1000 rounds a year.
It sounds like for what you are trying to do, the 308 or 30-06 would suit you better than the 300WM.
When I am faced with a similar situation, I will always go the route of the 308. The 308 is never a poor choice unless you're hunting brown bears or african game.
Long range shooting is a process that ends with a result. Once you start to focus on the result (how bad your last shot was, how big the group is going to be, what your buck will score, what your match score is, what place you are in...) then you loose the capacity to focus on the process.