He enlisted in either March 1967 or in July 1967 and served at Fort Knox in the 100th Division (Training) of the U.S. Army Reserve until August 15, 1967. He was assigned Selective Service number 15-131-42-44.
Addison M. McConnell, Jr., reported for duty as instructed, possibly in April or May, 1967. After that, though, things get murky. Convoluted explanations, obfuscations, outright denials, ("Senator McConnell had polio as a child and he never served in armed services.” said Robert Steurer, press secretary for McConnell in Washington. “I am not aware of him ever serving.”)
In Richard Flahavan's extract he writes "the U.S. Army ordered him to undergo an Armed Forces Physical Examination which he did July 9, 1967. Apparently, he did not pass because he was released from the U.S. Army Reserve August 15, 1967."
A new member of the 100th Army Reserve Unit in Basic Training at Ft. Knox was arrested in the barracks for sodomy. The guy is getting out of the military and the excuse will be due to an illness, an eye disorder. The Major then told a joke about it saying, "I guess the guy couldn't see the difference between guys and girls."
Much more-older story:
https://www.dailykos.com/stories/2013/2/24/1189537/-The-Strange-and-Possibly-Sordid-Story-of-Mitch-McConnell-s-Military-Service