The 30-06 is fine with a quality bullet and shot placement. Heck, I would use my 257 Weatherby on them, with the right bullet and a good shot placement, but a larger caliber is preferable.
I would just use the 30-06 focusing on a premium bullet. Check into the new Winchester XP3 bullet. I have read really good things on it and they claim it will expand quickly and penatrate deeply. Supposed to be good for both Deer, Elk and Moose.
I have a 270 WSM that I use on deer, but even then I usually load toward the bottom of the scale, closer to standard 270 velocity. If I was using it on Elk or Moose, I would load towards the top end with a premium bullet.
I just think too many people focus way too much on another caliber instead of focusing on a proper bullet choice for the caliber they already have that will do the job.