Got to throw my vote in for re-chambering or re-barrelling to 35 Whelen as it is effective & factory brass & ammo is easily available. It is about the minimum to enable you to jump to a significantly bigger bullet, caliber & weight wise. The 338 is only a half step.:) Yes, I am biased!
Further, I have some experience with the Whelen & it has been easy to get along with and never disappointed me.
Practical side of me needs to comment that your '06, with any 165-180 gr bullet, including plain old Core Lokts, has proven itself to be seriously effective on everything up through, and including moose.
