Your 308 when loaded with the right ammo can easily kill whitetails at 800 yards and elk at 500.
But if what you want is a rifle that will kill elk at longrange, then you need to be looking at the magnums e.g. 7rem mag, 300win mag, 300RUM etc. An alternative to the Sendero (which is a good suggestion), you could go with a much less expensive Rem700 SPS. You can pick one up new for <$500, then change the stock for an inexpensive laminate ($100) and bed it ($100 if a gunsmith does it), install muzzle brake (~$150). This leaves you with $1300 for bases, rings and optic.
Rather than spend the entire $1300 on optics, consider leaving some cash in the budget for reloading gear. You can get started reloading for ~$500.
Although my nephew's rifle is not a magnum, it is a great example of what you can do: Nephew's Transformed Rifle!