To MNoirot,
Dunno what kind of .308 you are shooting that lets you down past 500yds. I'm thinking you'll just see those results magnified by going to a larger recoiling, heavier powder-charged cartridge.
If what you're looking for is accuracy and precision, the ticket is lesser caliber cartridges with super-high ballistic coeficient, and case capacity at optimum to deliver the velocity you need at distance you anticipate.
One cartridge comes to mind, as a great 1000yd performer; the 6mm XC designed by David Tubb. The ctg is basically a 6mm International with AI'd shoulder. I have a 6mm/22-250 AI which is .001 longer in case dimension, otherwise identical. This cartridge will deliver very high BC match bullets with minimum recoil and also with substantially better barrel life, than items like the .243, and 6.5/284.
As far as 1000yd range results go, in 2008 or 9, Dominion Rifle Club of Canada posted results of .223rem fired with 90gr JLK and Sierra Matchking bullets in a very in-depth test. Groups averaged a vertical spread of about 2" in 15 or 20 shot groups, all fired from bolt gun w/1:6.5 twist barrel at 1000yds. The potential is there. A high BC bullet with minimal recoil can deliver the goods.
To get consistent accuracy and precision, really have to have all the variables dialed-in. I have seen scope mounts play a big role in my efforts, and even the ability to set precise inch pound torque values on action and sight mount system has made a difference. Then there is the shooter's ability to read the wind and mirage...
Maybe all you need to get your .308win performing as you'd like is a canted scope base and quality tracking optic with lots of elevation adjustment? Lots of variables to deal with, but rather doubtful going to a .30 magnum or other LongRange performer is going to bring you what you want until you solve the variables at work with your .308...
Of course, you don't really describe your rifle or the ammunition you're using. You just seem to want a purchase solution. Buy something like a Sako TRG-42 and maybe your results will change if you're having rifle problems. If the problems are of your own making, then the new rifle won't likely do any better; in fact due to much heavier recoil, may deliver even worse results. Best to KNOW how all the variables interact and have experience isolating and solving them.
Wish you the best of luck!