I believe it was Bill who said to Eddie, "We always figured you'd hit it big writing songs for Broadway" (something to that effect), so I would safely assume he worked not only as a piano player but also a songwriter.

Since many of the characters, especially the waitress, seemed to know "the song" which he played often and which was being played even after he was killed, it may also be safe to assume "the song" was a hit and was well-known, if not even being a "one-hit wonder." The hit song, however, would lead me to also guess that perhaps Eddie had a string of minor hits and maybe even penned a few minor scores for off-Broadway shows.

Add any compensation from his musical career to any money he received from his military service. Also, let's suspend disbelief and imagine he even had a trust fund. Or perhaps he had a relative that passed away and left him a handsome estate.

Personally, I like to imagine that Eddie pressed the Chinaman Button and in a strange twist of fate, he received a large sum of money from a wealthy cousin in South Africa who suddenly "passed away."

But that's just me of course.