With a .500 you have the big end taken care of.

I would go for a 9.3x74R or the .375 Flanged for the second.

Both will double for the .500 if needed for elephant and buffalo.

The 9.3 might not be technically legal in some countries for these game however.

Both have similar power, the .375 having a small edge. The recoil of both will be quite manageable even for persons finding bigger bore difficult.

I like the .375 Flanged as I think it is a better cartridge.

However loaded ammunition can be found for the 9.3 throughout Europe and probably South Africa and Namibia.

The 9.3 is extensively used for boar hunting and would also be excellent for moose.

I would go for the 9.3x74R which would be excellent paired with the .500 NE. I have a 9.3x74R and haven't found it lacking including good shots on water buffalo and other similar sized game and smaller ones too.

Another choice would be the 8x75RS. It is also offered in Heym's double rifles as well. A 8mm case based on the 74mm Rimmed case. Perhaps offering the ability of somewhat longer ranging. Given any limitations a double rifle may have of course. A 220 gr projectile in the 8x75RS would also not be lacking for moose, red deer and boar. And African plains game a little bit further out.

I can't see it listed on any site but it used to be.