Use the link below to show all soldiers listed in the 32nd Tennessee. There is a Joseph H. Smith listed as being in Co. G of the 32nd. He is also listed in the book Tennesseans in the Civil War, part 2 as being in Co. G of the 32nd. Since most of the 32nd came from the areas around Marshall, Lincoln, Giles,Franklin, Lawrence and Williamson counties this is likely your man.
Be patient as it will take some time for the list to load.
http://www.itd.nps.gov/cwss/Soldier_Results.cfm?maxcount=2018&start=1&unitcode=CTN0032RI