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.

On a completely non-Grateful Dead related note, there is a silly song about alligators which goes:

The alligator is my friend
He can be your friend too
You can wear him on your head
Or wear him on your shoe

Can be your friend can be your friend... too!

The accompanying hand motions (what, you thought this wasn't a multi-media performance piece?) include pointing at the head and feet, and opening and closing fully-extended arms, one over the other, like the jaws of an alligator, for each mention of the word "alligator." In particular, on the second singing of "alligator," the arm-jaws are opened wide for the whole long extended first syllable, because that is silly and fun, of course.
And very lastly, the word "alligator" is a highlight of Kermit the Frog's first scene in The Muppet Movie (1979). "Read my lips," the Muppet says, and his face enunciates every bit as much as his voice: "AL-LEY-GAY-TOR." ♥ Yay!

Last updated 6 April 2012 because more alligator trivia is all in good fun, and if it can include Muppets, then so much the better.