"I didn't say his middle initial was S....I said his son's was."

I knew that as soon as I hit "Post."

So here are the other William or W. Fullers:

Private W. E. Fuller, Company A, 7th MS Cavalry
2nd Lt. W.G. Fuller, company K, 3rd MS Infantry (State Troops)
2nd Lt. W.G. Fuller Company K, 3rd Regt., (Militia) Minute Men
Private W.W. Fuller, Company E, 39th Regt. MS Infantry
Private William Fuller, Cook's Company, (MS Horse Artillary)
Private William Fuller, Company F, 1st Regt., Light Artillary (Bradford's Battery)
Private William Fuller, Hoskins battery; Brookhaven Light Artillary
Private William Fuller, Company M, Wirt Adams' Cavalry

Again, good look in your search. Stop back again and let us know what you found out.