I found that a song called "Lay Something on the Bar (Besides Your Elbows)" was recorded by Louis Jordan and appears on CD#7 of his 9-CD set called "Let the Good Times Roll: The Complete Decca Recordings 1938-54" (Bear Family) where the authorship is credited to "(Austin/Smith)." That might be William Austin and Charlie Smith, who played piano and drums, respectively, with Jordan. Jordan played clarinet and sax and did vocals, and his style is described by AMG as "East Coast Blues, Jump Blues, Swing, and Urban Blues."

I have also found a song called, "Lay Something on My Bed Besides a Blanket" recorded by Charly (Charlotte) McClain on her album "Anthology" in 1999. This one is apparently "country." It is attributed to "(Hogan/Scaife/Scaife)." Maybe it has nothing in common with the first song but the similarity of the title.

Above are the only 2 songs indexed in AMG that start with "Lay Something…" or "Put Something…"

When I put "besides your elbows" into Google, I found that Jerry Lewis (the comedian) recorded a single called "Lay Something on the Table (Besides Your Elbows)" in 1951. However, another site says the word in the title is "Bar," not "Table."

Also, some sites, referring to Jordan's recording, say "Elbow," not "Elbows."

The title of this song is certainly intriguing.