A .30-06 shooting 180 grain Remington CoreLokt bullets and a 4x scope (or a 3.5-10x 40 mm scope) will work fine. Frankly, if you use a rifle such as this one, other factors are going to be more influential over whether you kill an elk or don't kill an elk. Can you find elk? Can you get close to the elk (close meaning can you approach the elk from the 1/2 mile to mile distance at which you first spot them to within whatever range you are capable of shooting at)? Can you shoot straight and accurate when you get to the elk? My guess is that the success of many more elk hunts hinge on these first three questions than what cartridge you use.
Of course other cartridges work also. You could use a .338 winchester magnum for deer hunting and elk hunting. More expense for cartridges. More recoil. Would it work? More than likely. Are you going to be super accurate with the .338 win mag? are you going to shoot hundreds of rounds at the range practicing with the .338 win mag? Probably not. But if you get within 200 yards of an elk and you just shoot 4 MOA accuracy, you should be able to land a first killing shot on the elk. Again, the trick is getting in a position to shoot the elk -- finding elk and then closing in on the elk in a timely manner. A lot of cartridges will work. Probably not a .22 LR. Probably not a .30-30. But that still leaves a lot of viable candidates -- and the .30-06 is a pretty good candidate. Lots of elk are killed every year with .30-06.
Taking a back-up on a distant hunt is a good idea, one someone mentioned above.