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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).
It suprised me but the 7mm Mauser ammo is available at Wally World as is the 44-40. But if you can't find ammo you like try Gene Sear's Supply in El Reno, they have a website too and your local gun shop can order from them. Have you ever seen over 100 Browning Belgiums in one spot in all the grades made? Or an original Winchester 44-40 saddle rifle? They have them. Sometime I go there to just drool!