Recruits came on roll at different times during the war. Since the service file doesn't show Cunningham on the original roll for August 5, 1861, I don't know why we could conclude he joined at that time.
Another man named Franklin L. Cunningham also appears in service files for Co. "D", 1st Tennessee Cavalry. This man applied for a pension from Rhea County and apparently lived there in 1860. Company "D" of Carter's regiment was recruited from Rhea, Roane and Bledsoe Counties. Any relation?