On the coldest night of the winter, Mazou Mounkaila was sleeping under an overpass in the Bronx when the ambulance crew arrived.

The wind chill was minus 4 degrees. Paramedics and homeless-outreach workers told Mr. Mounkaila he had to go either to a shelter or a hospital. Mr. Mounkaila, a courtly former warehouse manager from the West African nation of Niger who has been homeless for about a decade, declined to do either.

But he had no choice. The police showed up. “To my surprise,” Mr. Mounkaila said, “they handcuff me.” He spent the next 104 days at Jacobi Medical Center in the Bronx being treated for schizophrenia.

The mayor’s directive calls for the police, paramedics and groups that work with homeless New Yorkers to send people to hospitals when mental illness leaves them unable to “meet basic living needs,” even if they’re not threatening to hurt themselves or others.

It has met with criticism and a legal challenge. But one social-service agency in the Bronx that has been sending people to hospitals says the policy is yielding encouraging results, thanks largely to more diligent and longer-term hospital treatment.

As American cities struggle to turn back a rising tide of homelessness, New York is part of a broader movement to reconsider longer psychiatric hospital stays, half a century after mental institutions that had become brutal warehouses of humanity were emptied but never replaced with a coherent system of care.

Alex V. Barnard, a New York University sociologist who studies psychiatric hospitalization, said that Mr. Adams was one of several prominent Democrats, including leaders in California and Oregon, seeking to “sort of reframe coercion as compassion.”

Relatively few people in New York have been hospitalized under the “basic needs” standard, though the city declined to say how many.

But BronxWorks, the agency with the city outreach contract in the Bronx, has sent nine clients to hospitals, including Mr. Mounkaila. Most have either moved into permanent housing or are on track to do so.

One woman who was hospitalized was in her late 70s and had been homeless for so long that her campsite on White Plains Road shows up in Google Street View images going back to 2016.

BronxWorks said that city hospitals, which have been criticized for quickly discharging homeless psychiatric patients without helping them, now often keep people long enough to give them meaningful treatment and get them ready for housing.

“These are severely mentally ill people we’ve been chasing after for years,” said Scott Auwarter, BronxWorks’s assistant executive director. “Something’s changed out there. It’s working.”

The people sent to hospitals under Mr. Adams’s directive have typically not been those whom many New Yorkers are most worried about: people who are violent. But city officials say the directive is aimed at getting help for those who fall through the cracks over and over, whether or not they are a threat.

“This is really about focusing on a fairly small population of people that we know,” said Brian Stettin, the mayor’s senior adviser for severe mental illness, “because they’re kind of stuck in the revolving door of the system.” People threatening violence are still subject to hospitalization or arrest if the police happen to be on the scene.

Mr. Mounkaila was one of six hospitalized BronxWorks clients who were on a so-called Top 50 watch list of chronically homeless people with mental illness whom the city considers particularly resistant to intervention. Being on the list does not mean someone is targeted for involuntary hospitalization, but it means that the system keeps an eye on them — at least in theory.

Last month, Mr. Adams seized on the death of Jordan Neely, a man on the Top 50 list who was acting erratically when he was fatally choked by a subway passenger, to re-emphasize the case for forced hospitalization.

“We must look at involuntary removal of those who cannot take care of their basic needs and are in danger to themselves,” he said. “It breaks my heart how Jordan lost his life.”

Some families of patients hospitalized by BronxWorks had helplessly watched loved ones deteriorate.

“I begged for help, I asked God to help me with my brother,” said Orlando Solano, who said his 67-year-old brother, a onetime doctor, started living in the streets, chased by paranoid delusions, about 15 years ago. “I even looked for some paramedics and told them the situation, but they told me that they couldn’t do anything unless my brother had a very serious crisis.”

His brother is at a nursing home where his mental and physical health are so improved he’s like “another person,” Mr. Solano said.

While a shortage of hospital psychiatric beds persists in New York, the state and city have recently opened about 80 beds where patients can get extended care for months.

“Nothing about a seven-to-14-day hospitalization is going to change the arc of the life of somebody who’s homeless and has schizophrenia,” said Mitchell Katz, head of the city’s public hospitals.

Outreach groups have seldom invoked the power to have clients removed, because their work — a delicate dance of getting someone to accept help and move indoors — relies on trust. If the hospital would not help the patient, damaging that trust did not seem worth it.

“If you’re going to forcibly take away somebody’s rights and take them to the hospital against their will, you want to make sure you’re not making the situation worse,” Mr. Auwarter said.

BronxWorks sent its first person to the hospital under the mayor’s directive on Christmas night, when the wind chill was 12 degrees. Outreach workers at an elevated subway station approached a woman in her 60s who had refused to talk to them for five years. Her legs were covered with ulcers and so swollen she could not get socks on, and one foot was bare. She screamed when the police arrived, but she got in the ambulance.

At Jacobi, as treatment and medication took hold, the woman warmed, said Olivia Cooley, BronxWorks’s clinical coordinator for outreach: “She went to saying, ‘When are you going to visit again and see me next?’” After five months in a new state-run bed at Manhattan Psychiatric Center, the woman was accepted into permanent supportive housing this month.

Five clients BronxWorks removed are in or headed for permanent housing. Three remain in hospitals or nursing homes. The last is on the street, because, BronxWorks said, a city hospital, Lincoln Medical Center, discharged him the same day, without notifying BronxWorks.

Juan Rivera, BronxWorks’s outreach director, said he was seeing “more collaboration and more buy-in” from hospital staff who in the past sometimes failed to consider patients’ psychiatric histories when discharging them.

Mr. Stettin said that a push led by the Adams administration had “made a difference in how we’re handling these cases on the ground.”

The pending legal challenge to the mayor’s directive argues against having the police make decisions about whom to send to the hospital. The lawsuit does not address hospitalizations like those initiated by BronxWorks.

Matt Kudish, chief executive of the National Alliance on Mental Illness of New York City, a plaintiff in the case, said that while forced short hospital stays typically do not have good outcomes, it was “great to hear” from a reporter that BronxWorks had seen clients move from the street to housing under the directive.

Mr. Rivera said involuntary hospitalization was always a last resort, but sometimes essential.

“It’s our job to keep people safe,” he said. “We can leave someone with wounds that clearly look infected and is sitting in feces and urine. Do you think they have a right to stay there? Maybe. But do we have a responsibility as social service providers, and social workers, and ultimately as human beings to look out for this person, because if we don’t, who’s going to do it?”

He added that, too often, they are called to identify clients who died. According to the city, exposure to cold killed 12 unsheltered people from mid-2021 to mid-2022 — the second leading “external” cause of death after overdoses.

While most people BronxWorks sent to hospitals were not violent, one woman had brandished a makeshift weapon at outreach workers. She responded quickly to medication and has moved into supportive housing.

While Mr. Mounkaila was at Jacobi, a city-run hospital, his younger brother, a nurse practitioner, visited him. For years, the brother said, Mr. Mounkaila had angrily accused him of spying for the F.B.I. and C.I.A. At the hospital, Mr. Mounkaila said, “We embrace each other.”

Mr. Mounkaila had mixed feelings about his hospital stint. He liked getting three meals a day. He took his first shower in 10 months. He reunited with a daughter he hadn’t seen in eight years. But he was being medicated against his will, and he was not free.

“I’m like in prison,” he said one April afternoon in a bright visiting room decorated with art by patients, including a menagerie of African animals he drew.

On May 18, he moved into an apartment-style shelter run by BronxWorks, which is working to find him supportive housing. He takes his antipsychotic medication — not because he thinks he needs it, but because he says BronxWorks would be upset if he stopped.

“Life is good,” Mr. Mounkaila said outside his shelter. “This place is better than living in the streets.”

Joseph Goldstein and Raúl Vilchis contributed reporting.