Came across this in my reading...
Veterans of the Civil War often visited the battle sites in Atlanta. In the first week of Feb 1889 General Grenville Dodge met his former division commander, General John Fuller in Atlanta and rode out with a sightseeing group of veterans to a site of contested ground. According to a reporter with them...
"The woods remained untouched since the day the General (General lMcPherson) yielded his life in the lonely thicket. A tall pine tree stands near. Through its rough bark are several deep cuts, where relic-hunters have searched for bullets. Even the corners of the granite base (of the vertical cannon) have been chopped off as mementos of the dead hero.
So it seems relic hunting started not long after the end of the civil war. I have even read that civilian scavengers went among the dead after the battle robbing the bodies of items including guns and swords.