Robert A. Allen, who did JJ1's soundtrack, also did music for some of Epic's early point and click adventure games such as Dare to Dream.

Although, I can't find so much of a demo of these games, so I can't say much for the music itself.

EDIT- I did just find a demo of Dare to dream here, but the music is more akin to American McGee's Alice than his music for the JJ series (Robert also composed the boss music for JJ2).