You have a problem with the identity of George Hill, as he does not appear with the family on either the 1850 or 1860 census. Children of James and Lucinda Hill are Henry, Caroline, Elisha, Richard George and James Asa Hill.

The closest that I find for Henry H or H H Hill would be a man who enlisted in Columbus GA in Co "C", 2nd Georgia Regt. He was discharged in late 1861, probably due to ill health. There is another H H Hill from Tunnel Hill GA who enlisted at LaFayette GA and served honorably throughout the war. There is yet another H H Hill (whose first name appears to be Horace) who served in the 3rd Confederate Cavalry from Murray County GA.

Can't imagine that a man from De Kalb County AL went to Columbus GA to volunteer before the smoke cleared at Fort Sumter and then went over to the other side.