As with any historical novel, research is the key. You are on the right track in doing a lot of reading.
They would be able to dry the meat, though that means some time with a fire. A hunter could cut a few pieces, dry a few pounds of meat that he could carry, and then move on in his travels.
One thing to be aware of is geography. You mention this character killing a deer in the plains. That sounds more like our great plains region, and those are flat grasslands with mule deer far more prevalent than white-tails. He'd be outside of the United States proper and in Spanish or French territory at the time until the Louisiana Purchase.
You can find grassland east of the Mississippi River, as Illinois is known as the prairie state, but "plains" generally means something west of the 98th meridian. Here is a link with a map showing it.
https://www.earthmagazine.org/articl...00th-meridian/ Historically, when you get west of that 98th line, it means grasslands, dry climate, etc. It's a different world from the forest regions east of it where whitetails typically live.
For your research, look up Walter Prescott Webb. He's an historian you might find helpful.
https://faculty.weber.edu/kmackay/wa...great_plai.htm