I'd be most concerned about the type of hunting rifle selected, rather than which particular cartridge. This is after decades of hunting. There are many, many answers.
Here on this forum, we might be a little prejudiced in favor of rifles built specifically for long-range hunting/shooting, but for a general-purpose hunting rifle it's hard to beat the basic "American Sporter" type rifle. A mid-weight, bolt action repeater in any of a dozen different cartridges, topped with quality glass.
So - is your 308 a "general purpose" type hunting rifle or is it a more specialized precision rifle? If it's a good general purpose type hunting rifle, then you're already set! Sure you can bump up to something bigger. But why?
Let me use my favorite "all-around" rifle as an illustration of what I have found to be about ideal, for me and my hunting. Before I get into what it is, let's see what it has done in recent years:
2016, I used one rifle/scope/bullet combo to take:
Antelope at 245 yards
Mule deer at 140 yards
Black bear at 325 yards
Cow elk at 338 yards
Then a few months later I drove to Alaska where I shot a wolf and a grizzly, with the same rifle. I did bump up to a heavier bullet. The grizzly was taken at under 40 yards, the wolf at about 250 yards.
All that, one rifle, one scope, and only two loads. In truth, I could have done it all with one load, no problem. The rifle is reasonably light. Handles well. Has better than MOA accuracy. Is extremely reliable in all sorts of conditions. It's not magic, it's just a good old normal north American type hunting rifle. Any number of different variations on the theme would do as well.
Mine is a 30-06 Remington 700 CDL which had an older 2-7x Redfield on it for all that hunting. Now it has a 6x Leupold. The loads were:
165 gr Nosler Ballistic Tip at 2970 fps via H4350
200 gr Nosler Partition at 2600 fps via H4350
I honestly couldn't ask for better results afield. It has become my "go-to" hunting rifle for pretty much everything. Simple. Reliable. Accurate. Adequately powerful.
Now - go figure out which you like best - Remington, Ruger, Savage, Weatherby, Winchester... Whatever. The rifle you like best in any number of good cartridges will do just fine. I like your 7mm Rem mag idea, hunted with one of those for some years. It's a good all-arounder, with about as much recoil as I can shoot well from a typical sporter rifle. Somewhere along the line, I decided to drop the 7 mag and go back to my old faithful 30-06, but that doesn't mean there was anything wrong with the 7mm cartridge. Either one, or many others, work just fine.
Regards, Guy