All discharged Veterans were encouraged to filed copies of DD214 with the County Recorders Office in the County they returned to live in after the war. Some states/Counties have them filed with the Family Court. Check all the County Offices/Courts in all the places he lived since the war. Some states offered Veterans Benefits separate from Federal so check with State Veterans Affairs Office either at the state level or counties he lived in. If he went to college after his discharge see if transcripts or records have any educational G.I.Benefits used. Look back on Property records as some states like California had home loans available. His service number maybe on any of these records.
Fold 3 and Ancestry have Military records online outside chance may find something there. Your local library may have one of them to use for free or a LDS Family History Center.
Also do a newspaper search of his home town before he was drafted there maybe an announcement of his entering the service or when he completed basic training. Good Luck.
Lastly talk to family...your mothers too...they may have memories of "Army stories" told at family gatherings early in his marriage or letters written between parents and siblings and himself while serving.
Edit: Also look for the spelling Keen as that is what the spelling is in the 1940 census. Also I find no Military databases on Ancestry referencing him.