Welcome Taylor. It is a comfort to have something that no doubt gave your father hours of joy afield.
Unfortunately, that serial number is not part of the existing Hunter Arms records, but should be from about 1918 which is when the lowest grade hammer gun designation changed from 'F' to 'H'.

There is quite a lot of information on the LCSCA Home Page, esp. under 'Shotguns' and the FAQs
https://lcsca.clubexpress.com/

The Hunter Arms Co. did not mark the gauge (bore size) on the receiver or barrels of Smith guns.
Rotate the top lever to the right, open the gun and measure across the chamber. DO NOT include the recess for the rim of the shotshell.
12 gauge is 0.810" or about 20.6 mm
16 gauge is 0.745" or about 19.0 mm
20 gauge is 0.698" or about 17.7 mm

The gun is likely due for a good check and clean by a double gun smith. In what part of the country are you?