KISS principal for harvesting game is best IMHO. That's why I like single shot Ruger Number # 1's for the lower 48 an my old Pre-64 Winchester 70 in 375H&H with just fixed sights for places where I ain't high on the food chain.........:o)
We had a Stevens Crackshot .22 for rabbits and close in varmits, I still have it. An Iver Johnson single shot 16ga, it was my dad's first gun and I still have it. A 30-06, M1903 Springfield that an uncle used for years but gave to my dad for deer, elk, antelope and varmits at long range. The favorite we had is still in the closet here at the house, a Winchester Model 12 with 30 inch FC. Dad bought it the September after I was born.
My first rifle that was my "very own" to have was a Western Field single shot .22 rifle. I bemoaned it being a single shot, but dad told me that I would learn to be a better shot that way. He was right on target. Many buddies in school had pumps, bolt actions, lever actions and semi-automatics, but I got just as many jack rabbits as they did.
A couple of years ago my son came across a 30-06 Harrington and Richardson Handi Rifle. Talk about recoil!! The rifle must weigh in at about 4 1/2 to 5 lbs with scope. I could handle your 375 any day compared to this rifle. There is hardly any drop to the stock, so the recoil gets shoved straight back. Wow! Not a good choice for only one rifle. Glad we have several others to choose from, this is only for the stout hearted.
As far as answering the question. I would guess it depends on where you hunt. I have a Pre-64 Model 70 in .243 with a varmit barrel and 3X9 Redfield Widefield Scope that is just the ticket for deer in the hills of Central Texas (as long as I didn't have to pack it too far). Here in Central Kentucky my son and son-in-law each have Marlin 336's in the venerable 30-30 and I still use the Springfield 30-06 that has been in the family for over 55 years. Hunting deer on Ft Knox requires use of either a shotgun and slug or black powder. For that I have a Remington 870 with open sights and a Thompson Center Hawkin in .50 cal.
The original poster to this thread was talking about going on the cheap. In that case I guess for close ranges such as he would more than likely find in Louisiana one could almost get by with a shotgun and two barrels, one for slugs and one for birds/bunnies.
I find myself rambling - need to go get some honey do taken care of and a Stock Car race is in need of watching.