I have a Les Baer .308 Semi-Auto Sniper
with a 20-inch barrel that I use for precision tactical rifle competition.
One of the targets that is sometimes used is at 818 meters (894 yards) at the range where I shoot. The problem with shooting that distance is not so much the capability of the rifle or round, but the crosswind conditions encountered over that distance.
From the day you start shooting, keep a notebook with the ammunition used, distances shot and the environmental and wind conditions. Building your personal "dope" information for the rifle will be an invaluable asset for you.
If you want to shoot over 600 yards, depending upon the size of the target, I'd recommend a scope with a higher magnification than 15x. I use a U.S. Optics 3.8 - 22x scope, and at times wish it would go up a bit higher in magnification.
One of the things I have found to have a significant effect on accuracy is the concentricity of the ammunition. I bought a concentricity gage and check every round that I use. It's tedious, but if you can get (or verify) every bullet to be within 0.001 of zero, you will find that your accuracy will be improved.
If you can reload your own ammunition, that will help a great deal with concentricity. If you're buying .308, the Federal Match .308 ammunition quality control seems better than Remington, Winchester, or the standard Federal ammunition.