I would agree with the several folks that have advocated working-over your current gun rather than buy a new one. I'm in the same situation as you. I have a rifle and I'm not satisfied with the accuracy. I looked at what it would cost to buy a new rifle against custom gunsmith work. They were about the same. So, rather than buy another rifle, I've invested in a premium stock and will have the gun rebarrelled, square the action, lugs lapped, and the action bedded. If my trigger had not already been worked-over I would be doing that as well. This way I get a 'new' custom rifle
and should outperform any rifle out-of-the box even with factory ammo.
Your Savage is a great platform to start with. By re-working the gun the only thing you need to do is choose the caliber. I would also agree with the other folks here that for 600 yards you would want something heavier than a 204.
Let us know what you decide.