If you want versatility, especially for North America, the 30 caliber cartridges are good choices. Maybe the best. You already have a very capable dangerous game rifle, so I suggest your next rifle should be something you will shoot a lot with readily available, inexpensive ammunition. Many cartridges qualify for that category, - “mice to moose” I have some experience with moose, and most of the moose I have shot have been with the common and very capable .308 Winchester. Good statistics published by Sweden prove that there is no real “ killing power “ advantage to any particular moose cartridge in the following group: .308, 30-06, 300 win, 8x57js, 6.5x55, 7x64, etc. So I suggest you choose a rifle you like in .308 or .30-06. It will do everything your big double rifle is not well suited for, it will recoil less and make less muzzle blast than a magnum .300, will efficiently kill all non dangerous game at reasonable ranges, and will be inexpensive to buy common ammunition for. You are likely to practise more with a mild but capable cartridge. A well aimed hit with a .308 is infinitely better than a poor hit from a .375. I currently hunt with more than a dozen rifle cartridges that cover a wide range of calibers and that’s my observation and advice.
A good bolt action that fits you well, with a top quality scope of medium variable magnification, ( say 2-10x) chambered for .308 or .30-06 would be capable of hunting anything your double rifle isn’t appropriate for.