308's and Savages are always a good place to start. Benchrest rifles are a whole nother breed of cat especially at 600-800 yards. I guess it depends on your expectations.
You see the whole idea behind bench rest shooting is to shoot as small a group as possible. Now alot of this has to do with shooter skill, consistency and reading the elements as they change and compensating for those elements. At 600-800 yard you will be looking at wind direction, wind speed, mirage, lighting, temprature, pressure and humidity changes AND how they will affect where your next bullet will hit. For a target shooter, these things are of concern but not nearly on the same scale as a BR shooter. At 600 yards for a match shooter the 10 ring is 12". A shooter can shoot a 12" group and win the match with a 200/200 points. A BR shooter is looking for as tight a group as he can get. At 600-800 yards a good BR guy is looking for 2-3" groups on a bad day.
The 308 is capable of astounding accuracy when conditions are good. It falls short of 600-800 yard contention simply due to its slower qualities. This will cause more wind drift than a 300 WSM or RUM. If you missjudge the wind a bit, there goes your 800 yard 2" group. For a match shooter, that is ok. We can ussually judge well enough to keep a bullet in the 10 ring. then again the 10 ring is very forgiving. A 2" group is not forgiving.
Ask your self what your expectations are. You will see some excellent groups from your 308 at near and far ranges. The nice thing about a 308 is they are easy to load for, cheap to operate and barrels have a good long life. This all amounts to being able to test, experiemnt and practice. This is what will lead you to good groups. Not shooting the lates big boomer. Shooter skill and knowledge of how to use his tools is always the most important part of hitting what you want whether it be a 2" 800 yard group or an 8" target with the first round.
Good luck and welcome!