Instead of lookinf for a cartridge/calibre, examine perhaps what you need rifles for. The type of game animals and type of hunting.

You have a .375 which takes care of larger game, at the lower end of the choice range.

The 8x68S is a powerful magnum powered medium rifle.

As for the next choice?

If I looked at the ideal minimum sort of battery this is how I would do it. For my purposes at least.

1. Small game centrefire rifle. Accurate for head shots. Not over powered to reduce fur and meat damage. Scoped with a decent powered variable scope.

For me, this is a Mauser M03 .222 with Ziess Conquest 3.5-10x powered scope. Shooting rabbits, hares, foxes, roos, and smaller ferals.

2. Big game hunting rifle, bolt action. For buffalo and other large bovines and such.

I have a .375 and a .404 for this. Plus .450 double rifle.

3. Flat shooting powerful magnum rifle. For 'world' trophy hunting of plains game, deer, antelope etc. Scoped.

A .30-06 M98 and a 8x68S M03 fill this role.

4. Brush hunting rifle. For sambar deer in the forest, pigs in lignum etc.

My Browning lever action BLR .308 does this. A BLR is accurate and could easily be used for no.3 as well. If we had driven medium game hunts here, this rifle would also do well for that purpose. Fast shooting and reloading. Light, short and handy.

5. Lightweight mountain hunting rifle. Might be the same as no.3 if that rifle is light enough to be carried up and down mountains.

I don't have any lightweight rifles and make do with medium weioght rifle such as my Mauser 98 .30-06. Again the BLR .308 IS light and short and accurate and while unconventional as a mountain rifle would work as well as any of these customised Rem 600 customised rifles in .308.

Some people might choose also a lesser medium calibre, for easier shooting than a 8x68S, for smaller medium game species.

So what do you use your rifles for:
Monteria and driven game hunting of medium sized species, boar, red deer, other?
General stalking rifle for medium sized game, perhaps some mountainous terrain?
Other?

My guesses would be:

Driven and monteria type hunting. A 8x68S would do this.

But some driven game hunters have told me, a classic bolt action is too slow reloading for driven hunting. And so go for some of these more modern bolt actions, with inline bolts. Like the Blaser and other Euro makers. I am not that familiar with a lot of those rifle models. Obviously a medium calibre double rifle can also fill this role with two shots. A semi auto such as a BAR was once the ideal here in Australia prior to 1996.

Combined with a 8x68S, something like a 7x64 might be good in such a "faster" rifle, if such is needed.

For general stalking type hunting, again your 8x68S does this. But perhaps a lighter calibre is sometimes desired.

Again the 7x64 would be attractive to me. But by simple difference to what you have, and no particular chosen hunting need, I agree a Mannlicher in 6.5x54 M-S would be a cool addition. If wanting a better variety of projectiles and different rifles, yes I agree the 6.5x55.