Every day, people go to homeless shelters as a place of last resort. But some shelters, like the Marshall House in Hartford, Conn., are trying to help by turning people away.

At the shelter, run by the Salvation Army, intake staff members like Deborah Rucker are using an approach called “shelter diversion” or “problem solving.” Rather than supplying a bed and a blanket, Ms. Rucker helps people figure out if relatives or friends could help. She might make a phone call to a parent or a cousin to start that conversation. She can also buy gift cards for basic supplies, pay a utility bill, get new furniture or offer cash — small incentives to help make a doubling-up arrangement work.

On a cold October morning, Ms. Rucker sat with Margaret Guzman; her boyfriend, Alex; and their 2-year-old son, Jayden, in a drafty intake room at Marshall House. The family had been evicted and was running out of time to find another place to live. They feared they would have to spend the winter sleeping in their car. That poses a lethal risk for Jayden, who was born prematurely, does not yet walk or talk and requires an electrical outlet for a nebulizer for chronic lung disease.

Ms. Rucker said she had no beds for them, but then asked, “Do you have any family?” Around a small round table, she began to gently probe. What about Ms. Guzman’s mother, who lives nearby? No, her apartment was too small, and Ms. Guzman’s siblings already lived there. What about Alex’s mom? She has cancer, and it seemed too much to even ask. They kept talking while Ms. Rucker listened.

Ms. Rucker was using a diversion approach pioneered by the Cleveland Mediation Center nearly 20 years ago. The techniques are meant to resolve interpersonal conflicts that might also address homelessness. Studies have shown that millions of people double up with loved ones. But when relationships fray and the arrangements end, people turn to the streets or shelters. Even then, many people who are homeless stay in touch with relatives.

Doubling up is uncomfortable, stressful and in no way a long-term solution for people experiencing homelessness. But the shelter system itself can be very difficult to escape.