Here's another source of confusion. In addition to "Avery Island / April 1st," the melancholy instrumental in the middle of On Avery Island, "April 1st" is also the name of a relatively obscure track on the Hype City demo. This later song is probably better known by the alternate title "Swallowed," but if you take a look at the scan of the Hype City insert, you'll see it's labeled "April 1st." The differing names might be due to mislabeled mp3s of the song.
The significance of the date is unknown.

Also Known As
"Swallowed" (mislabel?)

Available
Hype City Soundtrack

Live
No known live performance

Lyrics

Everywhere I go
Swallowed up inside
(repeat a few times)