Pistol while deer hunting......

[Mr. 16 gauge]

Hello all;
Haven't been out hunting with the muzzleloader yet this year.....hopefully next weekend I can take off for a couple of days and fill one of my tags (haven't been out deer hunting at ALL yet this year).
Anyway, I was thinking that during our states muzzleloading season, small game season reopens again....rabbit, squirrel, grouse, and pheasant become legal targets. Shooting one with a .54 or .62 cal lead ball would be a bit messy.....so I thought about packing a smooth bore single shot loaded with bird shot. Anybody else ever do this? I'm just curious as to range, bore size, ect. Does anyone out there know of any good kits that might make a good summer project?
I currently carry a 1858 replica in .44 cal as well as my rifle, but I'm thinking about something I can use to take running shots at game. I already shoot a .62 trade musket, so if there is a pistol out there in this caliber, it might be ideal.
Thanks in advance.............