I think it depends on what part of the country you're in. How large are the moose in your area?
Here in ND, the meese
(Western sub-species) aren't big monsters like their Alaskan cousin. A cow will be around 500-750lbs and a mature bull will top out at 1500lb.
The Eastern is similar in size to the Western, maybe a touch smaller.
The .25-06 is on the bottom edge of being adequate for moose here in ND.
As already mentioned, use the heaviest bullet you can load (120gr) and use premium bullets like Barnes, Nosler Partition, Swift A-frame, etc.
If you keep your shots 150-200yds or under, you'll be fine energy-wise.
Go for a shoulder-bone breaking shot to anchor them as fast as possible.
If you do a lung shot (with any caliber) these guys will go a long ways...they've got tremendous lung capacity and can cover a lot of ground until they run out of oxygen.
I used the .25-06 with the 100gr Barnes X-bullet (my normal deer/elk load at the time) to harvest a 600lb cow with a 75yd shot.
Might've been a bit lighter than I should have used, but she dropped right where she stood with two broken shoulders.
If you're in an area with larger moose, I would definitely consider a larger bore.
One of my co-workers harvested his moose with the .308 and I know several others that have used the .270 and .30-06 succesfully. All of these will carry a bit more energy than the .25-06 and still have less recoil than your old 7mmMag.
Another good set of choices are any of the Mauser cartridges...6.5x55, 7x57, 8x57, etc but you'll likely not find these chamberings in many modern rifles.