I've seen two other rifles marked "A.H." and both were rather plain, swamped barrels, maple stocked, with good architecture. The othere rifles had butt plates, but were generally plain. I thought perhaps they were made in Kentucky, due to the strong similarities to many simple working rifles from the central part of that state that were iron mounted in whole or part, but I have not proven the connection.
The small bore tends to speak against the use of a much earlier barrel, because it would probably have a larger caliber if earlier. Many of these early percussion working rifles have been shortened one or more times at the breech, pulling the barrel back and reducing the light swamp at the breech end. Regardless, you have an interesting and nicely stocked working rifle, and it was great that you posted it for us to enjoy. Shelby Gallien