In my genealogy I don't have a Matthew Hale, but he is probably related to the following you had listed:

J.S.Hale = John S. Hale, b. 1842; d. 1922 Morganville, GA

James R.Hale = ?

S.J.Hale = Samuel Jackson, s. of William G. & Christinia Hale of Hooker, Dade Co, GA. b. 1826, TN; d. 1920, Hooker, GA.

S.M.Hale = Samuel, b. 1847, Dade Co, GA

These men were also listed in the 34th Reg., Co.F, GA Infantry along with my gr-grandfather, Charles Tittle.

This Hale line is incomplete, but can go back to Amon Hale, father of William G. Hale. Hope this helps in your search.