I would get something in .22 or 6mm for that "medium" distance (out to 300), for lower recoil and ammo cost.
Probably get a .204 ruger, .223 rem, or .22-250 rem, looking at CZ, Savage, Remington, Howa/Vanguard. I ran with a Vanguard for my most recent precision .223 rem. But a 6mm will buck the wind quite a bit better at 300 than any .22, including .22-250, so actually I'll revise and recommend that you look at a rifle in .243 win or even .260 rem - same makers. .308 is completely unnecessary amount of recoil and unduly slow bullet velocity, relative to say, a 6mm or .22.