As I recall, he started out with a .45 cal until he happened across Hatchet Jack's .50 Hawken. So he liked the big bores.
Now. If he had the opportunity to choose a modern cartridge, he would be inclined to choose the one that offers the biggest bullet, with the biggest thump downrange. In a Handi, that would include the .30-06, .35 Whelen, .45-70 or .450 Marlin. The '06 and .35 offer long range thump that the .45's cant, yet the biggest bullet he may be able to use would be a 220gr - 250gr. But, Jeremiah was from the era of big, slow moving bullets. As described in the movie "he was a man of sound mind and proper wit". The movie also portrayed him as a quiet, responsible and determined individual. He was a good tracker and knew where to go to find game. That in mind, he may not need beyond 200yd capabilities. It would seem that when he found something that worked, he stuck with it. So I think he would lean toward the Marlin.
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