The .308 is an excellent deer hunting round...especially in a compact rifle...my favorite being the Ruger M77II compact.
It has long been known for amazing accuracy, and it has nearly the power of the .30-06. It is fine for whitetail, pronghorn, sheep, etc. For elk or moose, I would also go with a 165 grain bullet in the Nosler partition or Swift A-frame line...Hornady also loads "Light Magnum" versions which are similar to the .300 H&H Magnum performance.
Its only weakness comes when you try to load heavy for caliber rounds, e.g. 180-220 grain bullets, because of the seating depth/case capacity...that is where the .30-06 and .300 Winchester / Weatherby Magnum has advantages because of larger case size.
Personally, I would avoid the 150 grain Winchester Silvertip...several bad experiences with that specific load...though I really like the Winchester Ballistic Silvertip...