I vote for Al Cohn as the tenor player. (Too bad they only show his hands during the solo).
As I recall, Scott was married to the featured singer Dorothy Collins at the time. Took home two healthy paychecks from that show...
Too jowly for Al at that time, I think, also looks about 10 or more years too old for him, and it sure doesn't sound like Al -- a kind of proto-Sam the Man Taylor vibe. Maybe Al Klink, who could sound like anyone on cue.
Boy, I used to think Dorothy Collins was cute, which is kind of pathetic in retrospect -- going for that scrubbed, nice-girl image and the rabbity slight-overbite. An extension of the teacher's crush, but what are you going to do at age 12 or so? IIRC, the Collins-Scott marriage ending in severe strife.
Truth time. I was there too. I had a crush on Dorothy Collins as a 10 year old in 1955. You're right, Larry - must have been a teacher's crush thing.