Probably not the song you want, but an interesting bit of trivia:

Hoagy Carmichael wrote (or perhaps started to write and never finished) a song called I LOVE YOU LIKE MY OLD FELT HAT.

An incomplete bit of sheet music, apparently in Hoagy's own handwriting, can be seen here, in the Hoagy Carmichael collection at Indiana University. It's the one crossed out at the top of the page. There are no lyrics.

It's listed at The Official Hoagy Carmichael official web site with no publication data, so I suppose that means it was never published (and perhaps never completed).

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I found this in several collections of tongue-twisters:

Of all the felt I ever felt,
I never felt a piece of felt
Which felt as fine as that felt felt,
When first I felt that felt hat's felt.

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Other than that, I'm stumped.