Student-loan forgiveness might seem out of reach for many of the 44 million people who have educational debt. But some of these borrowers may qualify for relief—if they know where to look.

Student-loan forgiveness has gotten somewhat of a bad rap in recent months, largely because of controversy surrounding the federal Public Service Loan Forgiveness program, which allows public servants with a certain type of federal student loans to have their debt discharged after 120 monthly payments.

The...