I hope JA did play the song, it has very nice lyrics, and anything from the Airplane is a must have. I can easily imagine Marty singing this one. Anyway:
Jefferson Airplane's first album was Jefferson Airplane Takes Off in 1966. It did include Blues from an Airplane and other songs. I am speculating that if Jefferson Airplane played a song like that, it would be relatively early. Every other subsequent major album does not include it, so it would have to be a live recording, or a studio version not released on any of their major albums.
Also what I know from searching the web, The Box Tops wrote The Letter which was released in 1967, and has the same lyrics as Give Me A Ticket For an Aeroplane.
I know this doesn't help, but that's all I know. You should definitely check out Takes Off even though it doesn't include your song, it is a great album that has songs that sound similar to that one, with the lyrics and such.
Anyway, welcome to the forum and come back often!!