A William Killgore [note spelling] served in Co. C of the 39th Tennessee [Bradford's]. This unit was originally the 31st Tenn. but due to another unit with the same designation, it was changed to the 39th in June 1863 when stationed at Vicksburg in Col. A. W. Reynold's brigade consisting of East Tennessee units.
William Killgore is also listed as having been wounded as cited in Lindsley's Annals.
As noted by LHares, only Co. H is shown as having been raised in Green Co.
Co. C of the 39th was raised in Jefferson Co.