Jasmine A. Braxton was working as a music-licensing intern when she decided she wanted to advocate for musicians as a lawyer. But when she graduated in 2014 from UC Hastings College of the Law, she was nervous about venturing solo into such a competitive specialty when legal jobs in general were so scarce (by March, just 60 percent of her class nationwide had found full-time jobs requiring bar passage).

So she turned to the Modest Means Incubator Program at California Lawyers for the Arts (CLA), which launched that summer. It is among dozens of similar programs created nationally since 2007 with two goals: To train new lawyers, and to address the “justice gap” by helping them serve people who cannot afford to hire a civil lawyer. In California alone, ten programs have launched since 2012, and two more aim to open by next year, according to the American Bar Association...