You have almost every thing you need for a 338-06. Nothing wrong with it, but I'd go .338 Winchester, or a short mag variant. Reason being I think thats where the heavier bullets start being useful. You can always load down, but pushing it hard to get what you want isn't alway feasible.
Although there is nothing wrong with a 338-06, I totally agree with Harperc on this. I personally think the 338 win mag is the best all-around elk cartridge ever made. period.
It can drive a 250 grain bullet fast enough to be a real hammer. You can shoot right through gamble oaks and other bushes without worrying if you have enough power (assuming the elk is right on the other side of bushes). As long as you put the bullet on a path that is going to hit the heart and/or lungs, it doesn't matter what the angle is... an elk shoulder will not stop a 250 grain bullet fired from a 338 win mag.
And you can build a 338 win mag into a hunting weight rifle without having excessive recoil. Although, at close range, I'm not sure an elk is going to know the difference between a 338 win mag, a 338wsm or a 338-06, 338 weatherby, 338 Edge, etc etc. They'll all get the job done with authority. Build what YOU want.