In Box of Rain, his published collection of Grateful Dead lyrics, Robert Hunter notes:

This was the first of my lyrics recorded by the Dead. I got paid two hundred fifty dollars from the record advance for Anthem of the Sun with which I bought a used car and headed north to Seattle, where I tried to make a living restringing beads from Goodwill for a friend's boutique. I made about five dollars at this occuption. The car broke down, which was okay, since I couldn't afford gas for it, so I hitchhiked back to S.F. and decided to hang in there with the Dead. Pigpen added lyrics of his own to "Alligator" and I've presented it here with his turns of phrase intact.

As noted above, the verse that begins "Ridin' down the river in an old canoe" is by Ron "Pigpen" McKernan, the Dead's first keyboardist.