Until seeing this discussion, the only time I had ever come across "Into the Air, Junior Birdmen" was in Tom Lehrer's spoken introduction to his song, It Makes A Fellow Proud To Be A Soldier:

I have only comparatively recently emerged from the United States Army so
that I'm now, of course, in the radioactive reserve. And, the usual
jokes about the Army aside, one of the many fine things one has to admit, is
the way that the Army has carried the American democratic ideal to its
logical conclusion, in the sense that not only do they prohibit
discrimination on the grounds of race, creed and color but also on the
grounds of ability. Be that as it may, some of you may recall the publicity
a few years ago attendant upon the Army's search for an
official Army song to be
the counterpart of the Navy's "Anchors Aweigh" and the Air Forces "Up In
The Air, Junior Birdman" and so on. I was in basic training at the time
and I recall our platoon sergeant, who was an unfrocked Marine.
(Actually the change of service had come as particular blow to him because it
meant that he had to memorize a new serial number which took up most of his
time.) Anyway, I recall this sergeant's informing me and my roommates
of this rather deplorable fact that the Army didn't have any official
... excuse me... didn't have no official song and suggested that we work
on this in our copious free time. Well, I submitted the following song which
is called "It Makes A Fellow Proud To Be A Soldier" which, I think,
demonstrates the proper spirit, you'll agree. However, the fact that it
did not win the contest I can ascribe only to blatant favoritism on part of
the judges.