If you plan on moving up to the "target/precision rifle category" and want to buy a rifle that will perform well straight out of the box without alteration, you'd be wise to pass on the Stevens. It's a hunting rifle and not suited to the task as such.
Depending on what your budget is, a Remington 700 SPS Varmint might be the way to go. I see that Savage isn't offering a .308 Winchester option in their 12FV series this year. Wow, a mistake on their part if they want to keep parity with Remington.
There are other Savage offerings that have the .308 option. You'll just have to decide what it is you want and need and go from there.
Has anyone had any experience with the new Savage Accu-stock? It has to be better than what they've had in the past. It wouldn't take much to be an improvement.
The "Golden Standard" for factory precision rifles has been the Remington 700P series. Buy it, lighten the trigger some, and away you go. The prices are starting to creep up on these so if $1,000 is out of your league then it's better to consider less expensive options.
Plan to spend as much if not more on QUALITY optics, base(s), and rings. If you can't see the target clearly or rely on the precision/consistency of the scope's windage & elevation adjustments, you'll never realize the true accuracy potential of either yourself or your rifle.
If I was starting out new and in your position, I wouldn't consider a .308 Win. but a .223 Remington. However, that's just me.