> 6.8 SPC
I'm going to vote for that one. Starline stocks brass for it, and there are still companies that load complete ammo.
The 6.8 is very close to the British 7mm Enfield "intermediate" cartridge that the Brits put a lot of work (including testing in actual combat) into. It was supposed to be a joint US/British thing, until the US Army announced they wouldn't even consider anything with less power than the .30-06 and the whole deal fell apart.
The 6.8 has decent ballistics, light recoil, and is good for medium game.
Another option would be the 6mm/.223 or 6x45. That's a wildcatted version of the .223 Remington. It was a popular bench rest cartridge for a while. The US Army tested it as a replacement for a 5.56x45. It was better in almost every test, but not better enough to make it worthwhile to change. (the same reason the .60 BMG was never adopted)
More recently, it has been extremely popular as a target cartridge in the Phillippines. There's also an impressive selection of appropriate target and hunting bullets in 6mm, and brass is dirt cheap.
Downside: might not be different enough from 5.56 to be worth your while, and I don't see any barrels for it out there, so you'd have to add "buy a reamer" and "profile a barrel blank" to the other costs of going to that caliber.