The latest rock-related death I can find for 1960 (even though he was more of a country singer) is Johnny Horton of Battle Of New Orleans fame, who died on 5th November that year - a bit closer, but still not quite there.
I seem to remember this question cropping up before without the answer being found.
I'm pretty old and was (and still am) a big rock'n'roll fan. I don't recall any big rock star dying on that day. The really memorable ones were the previously mentioned late Great Eddie Cochran, Buddy Holly, Gene Vincent, Johnny Burnette, Richie Valens and if he counts as a star after only one hit, JP Richardson (The Big Bopper) but none of them on that day.
As you will see this guy died in 1960 but no specific date given.
Kelsey Pharr - Died 1960 - ( Doo-Wop - R&B ) Was a member of The Delta Rhythm Boys (They did, "Just A-Sittin' And A-Rockin" and "For Sentimental Reasons").