I'm in a very similar boat, with the added complications of wanting suppress everything, wanting to stick to common chamberings, and being left-handed.
My deer rifle is also a .308, and on the top end, I have .375, though mine are in the .375 Ruger flavor. I also have a .300 WSM and 7mm-08, but those are on the chopping block since both of them would be a little difficult to suppress for various reasons. The 7mm-08 is also pretty much totally redundant since I have a .308. I've had a .338WM, but sold it since it really wouldn't do anything that a .300 or .375 can't do. Used to have a 30-06 as well, but didn't really need something in between the .308 and .300WSM.
Once I ditch the .300 WSM and 7mm-08, I'll be looking for either a .300WM or 7mm Rem Mag to fill the gap between .308 and .375 for me. My main use for the 7mm/.300WM would involve travel, either out west or Africa, which is the main reason I want to stick to a very common chambering. I'm leaning heavily toward the .300WM for several reasons, mainly because my primary goal is a harder hitting round at normal ranges, rather than a long-range round. I'll freely admit that i have no first-hand experience, but I'd expect the .300 to perform a bit better on African plains game than the 7mm. This is kind of splitting hairs here and I'm sure either would work just fine. If my primary goal was shooting deer-sized game at longer distances, I'd choose the 7mm Rem Mag instead.
Short version:
.308 - Deer and deer-sized critters at moderate ranges.
.300WM - Things bigger/tougher than deer, longer ranges.
.375 Ruger - Because I can/just in case I want to hunt mid-size dinosaurs.
These three each have their primary role, but enough overlap that one can be a backup for another in most cases. Realistically, if you can shoot the .375 well, I'm not totally sure that the gap between .308 and .375 truly needs filling, but getting another rifle is always fun, and something in between does provide some benefit.