.260 Rem and 7mm-08 are both ballistically superior to the .308, with less recoil and shooter fatigue. The downside is that neither are as readily available, or have as wide a selection, as .308. If you load your own ammo, it's not an issue, but if you don't it is something to consider.
You might think it to be a strange recommendation, but I suggest training your offhand shooting with a lever action rifle. The lock time is extremely slow compared to more modern bolt actions, and thus you must hold steadier for a longer period of time to be more accurate. Go for the Savage rifle that suits you, they make a good product. If you find a deal on a used short action savage and it already has the right bolt face, you may find it easier to swap barrels than track down the exact new model you want. The Savage 10/110 are the legos of the bolt action world.