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I tend to go along with the old adage that says: "Take enough gun."
Even well placed shots, if fired from marginal cartridges, don't do the job intended. There are many variables working against you, far more than you think.
Now, the 44-40 and 7mm Mauser are enough. Big slow moving slug for knock down punch inside 100 yds. (.44-40) and high velocity bullet for longer range, flatter shooting (7MM). They'll do for most situations.
I'm not one to generally recommend ammo that is so far down the popularity list, however. At least for most folks who are more into chickens than guns.
Something like a .44-40 (an old black powder revolver round) and the 7mm Mauser (a European rifle round) are wholly adequate for their purpose, but may not be readily available in all locales.
The popular equivalents would be the .44 Magnum/Special and something like a .270 WIN.
Both the .44 Mag/Sp and the .270 WIN are available in rifles and are very common.
(Were it me I'd prefer the 7mm-08, the modern duplicate of the 7mm Mauser. But again, not as common in all locales).
The 44-40 was a sorta magnum in the black powder days. Though hard to find loaded ammo, one can get casings from starline. Black powder is easy to reload, just deprime, resize, prime, and fill the case to the top with powder and shove a bullet on top.
I love shooting black powder, though I am not lucky enough to own a 44-40, I do load 44spl and 44mag in the black. I shoot a 45/70 buffalo rifle. It is not unusual for me to carry one of my 1851 navies converted to 38spl loaded in black. Surprising is that black powder cartridges have almost the same power as the smokeless in 38spl. Even my 12ga is loaded in the dark stuff.
I, too, love black powder and the vintage "charm" of the stuff. But for most people, its not gonna be a best choice. Simple and common place is better.
In the non +P loadings, a stiff charge of black powder should come close to orginal .38 Spl loads.
On a tangent, one of my favorite all around predator/farm rifle is the Marlin 1894C in .357 Mag/.38 Spl. It has a wide power range out to 100 yds and the ammo is as common as fleas on a dog. I recommend it highly.