I believe the name of that piece of music is Johnny Shiloh; I'm getting that from Persistence of Vision's World's Fair issue, where Buddy Baker's score for Great Moments with Mr. Lincoln is discussed. I believe this same piece of music is used currently in the American Adventure area loop and the Liberty Square area loop.
Also, I wish that Eric Paddon would chime in here, as I recall him saying that the Hall of Presidents underscore was used in the lobby of the attraction. He might have additional information.
Best,
wedroy1923