I don't know why you would replace the barrel and rechamber to a 308 The 30-06 will do anything the 308 can do and more. More importantly it will handle heavier bullets better.
If you are set on something else then you must consider ammunition availability of you reload then I agree with a 6.5-06 or even a 6mm-06. If not then a 25-06 may be what you want. the 280 remington is another good choice. Your options are limitless really.
If you want a shorter barrel 18 to 20 inches then the 308, 260, 243, or even a round based on the mauser like a 7mm mauser, 6mm remingtion or 257 roberts. Your options are limitless with the standard length action.
One thing to remember tho. When I was in the army I was sent to sniper school then was part of the rifle team. An instructor once told me that it really didn't matter what caliber of rifle one used as most rifles can shoot much further than the average shooter can accurately shoot them. I watched this instructer repeatedly hit a 36 inch bull at 400 meters with a ruger number 3 rifle in 47-70.
So my advice is pick your poison. I don't see where you can go wrong with whatever you decide
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