I really like this tune too.  Although it sounds like perhaps it was
written in the 1930's or 20's, it was actually written for the show.
The name is "Remembering You", by Roger Kellaway and Carroll O'Connor.

I believe that Roger Kellaway wrote the tune and Carroll O'Connor wrote
the lyrics.  The lyrics were never used in the show but here they are
if anyone is interested:

  Got a feelin' it's all over now - All over now, we're through.
  And tomorrow I'll be lonesome, Remembering You.
  Got a feelin' the sun will be gone - The day will be long and blue.
  And tomorrow I'll be cryin', Remembering You.
  There's a far away look in your eye when you try to pretend to me,
  That everything is the same as it used to be.
  I see it's all over now - All over now, we're through.
  And tomorrow I'll be startin' Remembering You.

I, too, would love to find this in _any_ note oriented format:
piano roll, MIDI file, sheet music (although that would be painful
to transcribe!).

Musically,

Ray Finch