Public Enemy's 1988 rap song "Don't Believe the Hype" contained the lyric "They still consider me a new jack." The term "new Jack" simply meant a newcomer, a novice, or a raw or aggressive youngster.
11 July 1989, Village Voice, pg. 34:
A new economic and psychic environment: New Jack City. New Jack City exists wherever there's a crack house, a back room stuffed with stolen VCRs and family jewelry, bleary-eyed young girls with bad breath willing to do the Unthinkable wuth the Anonymous....New Jack City exists in Harlem, the Bronx, Baltimore, Philadelphia, Detroit, Los Angeles, (etc.).
6 May 1990, New York Times, pg. 40:
The other film, "New Jack City," has had much less publicity. Filmed entirely in the city, it highlights rap music, drugs, police officers and other features of urban life.