There's also the patch on the left shoulder. It is this one:
Here is what this this web site says: The shoulder sleeve insignia was originally approved for the Supreme Headquarters, Allied Expeditionary Forces on 13 December 1944. It was redesignated for Headquarters, U.S. Forces, European Theater and revised to change the background color from black to dark blue on 2 August 1945. It was redesignated for the European Command on 23 June 1947. The insignia was redesignated for United States Army Europe on 7 November 1952. It was amended to include the specific shade of orange in the description on 22 July 1970. It was cancelled effective 16 July 2009. The insignia was reinstated for United States Army Europe on 7 July 2009, revoking the cancellation.
Once again - evidence of being in the occupation. Since you keep your "combat patch" when you go to another unit, having the 36th patch does't necessarily mean he was in the 36th when he left the service. It just means that he served in the 36th in combat. I'm afraid I don't know much about the AEF HQ - where they were after the war. I bet engineers were in big demand removing obstacles and mines. I bet a lot of German POWs did a lot of mineclearing.