Research reveals that Thomas Hall Hamilton after deserting from the 38th NYVI he also served later in the Civil War under the alias Thomas Hall (his mother's maiden name).

He was recruited by E. H. Hawkins June 14, 1864 and mustered-in to the 19th New York Independant Artillery Battery at Buffalo, NY. He was ordered to report to Elmira and was turned over to Colonel B. F. Tracy on June 23, 1864. Hawkins received $10 for delivering him to Elmira.
His enlistment papers gives the following information:
'Thomas Hall born in Lancaster, England aged 24 years and by occupation a Machinist, volunteered June 14, 1864 at Buffalo, NY. Described as having: 'blue eyes, light hair, fair complexion, 5ft 7-1/2 inches tall.' He was credited to the 30th State Senatorial District for bounty purposes. Thomas joined his Regiment during the siege of Petersburg, Virginia and fought in the following engagements:
Siege of Petersburg, through 2 April 1865
Mine Explosion, 30 July 1864
Weldon Road, August 18-2, 1864
Battle of Poplar Grove Church, 9 and 21 November 1864
Boydton Plank Road, 27-28 October 1864
Battle of Fort Stedman, 25 March 1865
Assault & Capture of Petersburg, 2 April 1865
Appomattox Campaign, 28 March-9 April 1865
Pursuit to Appomattox

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John Rector was wounded 12 May 1864 in the Battle of Spotsylvania Court House (Spotsylvania, VA). His battery and the others of the IX Corps were overrun by Lane's North Carolina Brigade. Its 4 cannon were silenced and the cannoneers shot. The 19th NY Light Battery suffered 16 casualties that day, its largest loss of the war - 7 killed, 10 wounded (3 later died). John was taken to Fredericksburg on 13 May, and on 25 May was transferred to Campbell Hospital No. 4306, Washington DC. Julia Wheelock's 'The Boys in White, The Experience of a Hospital Agent in and Around Washington' gives accounts of the wounded in Fredericksburg.