There’s a story that Glenn first tried to run as a Republican but got “dissed” by Rhodes’s (the then GOP guv & Ohio political powerhouse!) people, so he went over to cut a deal with the Rat side. The rest is history.
John Gizzi alluded to that in a 2016 column that Glenn had been a Republican who was being courted by legendary OH GOP Chairman Ray Bliss (presumably to take down the elderly freshman left-wing Dem nutter Sen. Stephen Young for 1964), but Glenn was successfully lobbied by JFK to come over to the Dems and challenge a very vulnerable Young intra-party (but was forced to drop out after he suffered his freak accident that damaged his ear, though he remained on the ballot).
https://www.newsmax.com/john-gizzi/hero-john-glenn-bad-timing-politics/2016/12/11/id/763354/