Sure it's enough, but there are better choices.
I agree I'd rather have a .30-06 but the .308 will work with a well placed shot.
And 06 will work with an ill placed shot?
There is almost no difference between the two when it comes to effects on game.
The added 100 Feet per second speed that the 30-06 has over the 308 in Factory loadings has minimal effects on drop out to 250 yards. Point blank Range (where your bullet drop will not be greater then the kill zone of the animal. Bullet drop (heck I sight in my 308 for 200 yards) would be an inch to three, based on where sight in your rifle and the kill zone on an elk is bout 5"
If you are looking to step up and gain more distance any of the short mags (300 WSM), short belted mags (300 Win Mag), or the full mags (300 H&H or WBY) would extend your range using the same bullet your 308Win is shooting only faster and there fore lowering the bullet drop over longer ranges as well as hitting with the 2,000 Foot pounds or more at those ranges. Of course the 300 WSM, and the 300 Win and H&H are only 200 -250 FPS faster than the 30-06 and would not extend you point blank range too much more.
Do not over think this. your 308 will work fine with in the limits. To change from a 308 to a 30-06 does not make sense for the added 100 feet per second. I have a loading manual that shows the 165 grain bullet going faster out of a 308 Win case than the same bullet and powder load leaving a 30-06 case.
So let's see with hand loads 308 Win can obtain faster speeds, The smaller case is more accurate, more efficient case, and can be made into lighter rifles( Depending on Action- single shot rifles will weight the same, long action bolt rifles will weight the same but short actions are lighter than standard actions) The 308 barrel lengths are 20 -22 inches and the 06 are 22- 24" Again heavier rifles. With the premium bullets you can go lighter and wring out all the ballistics of the smaller case while carring lighter ammo. All the things a mountain hunter wants. Not sure why you would want a 30-06 over a 308, but to each his own.