When Anders Osborne started his Send Me a Friend foundation last year, he had a simple goal: He wanted musicians in recovery from substance addiction to continue working as musicians. Calling on his own experiences of returning to music clubs when he was newly sober, he began assembling a network where “friends” in different cities could look out for players who were returning to their old haunts or going back on the road.

When Osborne went through that experience, he felt he was being for...