well if your not including a scope in the $500-$600, you have plenty of good choices.
as you mentioned and seem to prefer, .308 is a great round, probably my overall favorite do everything round.
the 30-06 is darn near as accurate, but you gain more variations in weights you can shoot with the 30-06. The CMP does sell us surplus 30-06, you can get 7.62x51 (.308 ) surplus from all sorts of countries of varying quality and price.
my top choice would be a remmy 700
and you cant go wrong with a .308 or 30-06.
<--- check there for the surplus 30-06