I'm certainly no bullet expert, but whenever I find these here in Ontario I've assumed they were rim-fire Henry Cartridges.
Here's a pic of some of the 'H' cartridges I've found. The small one on the right has the 'H' on it, it's just hard to see.
I don't have access to rimfire cartridge material where I could check on your rough dimensions as I have with my multiple reloading center fire manuals but with a case rim diameter of what appears to be something over a 1/4" I will guess a .32 caliber rimfire.
Winchester used the h head stamp back in the day, there is a couple of pics in my post above showing the H right on the winchester box.
Winchester honored Tyler Henry with the "H" headstamp for over 100 years, after introducing the Model 1866 Rifle.