The District’s homeless shelter in what once was D.C. General Hospital. (Bill O'Leary/The Washington Post)

D.C. officials are projecting a 10 percent increase in the number of families who will be homeless this winter. And with temperatures dropping and shelters still full, advocates worry that there will be, literally, nowhere for them to go.

“We are going into hypothermia season pretty much full,” said Sakina Thompson, interim administrator for the District’s Family Services Administration.

The family shelter that now operates year-round at the old D.C. General Hospital is still full, she said. Nineteen of 32 emergency units in community-based family shelters are full. And 21 families are still crowded into cheap hotel rooms, waiting for a spot to open up at D.C. General.

City officials, advocates and those who provide services to homeless residents will meet Tuesday to figure out this winter’s plan for dealing with the expected increase in homelessness. The city is projecting that 509 new families will become homeless this winter. They’re also projecting a slight increase, to 1,376, in homeless single men. The number of single women is expected to remain level at about 426.

“We’re looking very hard at the bottlenecks in the system,” Thompson said. “We’ve done a lot of work in the last year for homeless families on the front end, to keep them from coming into shelters. But once in, they’re staying longer. We need to move them out and into stable homes faster.”

Washington is one of the few cities in the country that guarantee that anyone without a home has a legal right to shelter whenever the temperature drops below freezing.

With a shrinking number of affordable apartments as the city gentrifies and fewer permanent subsidies available to help pay rent, city officials had hoped that a new program called Rapid Rehousing would move 60 families a month out of crowded shelters.

The idea behind Rapid Rehousing is to quickly get families out of shelters, where research has found it’s much harder to find work and much easier to sink into depression. The program provides short-term rental assistance for four months to two years while intensively working with recipients to them educations and better jobs.

But, Thompson said, the city has been able to move only about 27 to 35 families out of shelters and into apartments each month.

The holdup, she said, has been to find affordable apartments that also pass safety inspections.

“There are units that pass rent reasonableness guidelines but not the safety inspection,” Thompson said. “And we’re working hard not to put families in units that they can’t afford. We want to make their rent reasonable, not impossible. And that takes time.”

Jordan Love Smith, who has been staying at D.C. General since March, said she agreed in September to take Rapid Rehousing and signed a lease on an apartment. She said almost everyone she knows in the shelter is taking Rapid Rehousing now, after resisting it for months.

“But we’re all waiting on the inspections,” Smith said. “The neighborhood where the apartment is isn’t good, but it’s right on a bus line, and I’ll be able to manage. I ask them every single day when I can move out. And I’m still waiting.”

To speed the process, Thompson said, the Department of Human Services is looking for units itself and pre-certifying them, rather than waiting for homeless residents to find their own. DHS also is accelerating the movement of families into those units; providing services there, rather than in the shelter; and focusing those services on getting homeless residents work-ready or placing them in jobs.

In the city’s drive to keep people out of shelters in the first place, social workers at the Virginia Williams Family Resource Center offer food and rent assistance, Thompson said. She said that effort is a big reason why fewer families wound up in hotels last winter, compared with the winter before.

But Amber Harding, a lawyer with the Washington Legal Clinic for the Homeless, said that last year her group had to go to court to help 50 families get a place in the shelter. That was one-fifth of the families placed last winter.

“We agree with the city’s philosophy that people should be using the resources they have, but unfortunately, we’re seeing that the tighter they are on space, the more families have to ‘prove up’ their eligibility,” Harding said. “We had a client turned away from shelter last year who was told they’d provide her with a bus ticket to return to Philadelphia, where she’d just left her abuser. I think we can all agree that that’s not a safe place.”

Harding said she and other advocates also are concerned that the city has no backup plan if it runs out of hotel rooms. And, she said, she’s concerned that unaccompanied homeless youth, whose numbers have been growing, are not included in the winter plan.

Single homeless residents are, likewise, concerned about their future. At a community meeting Monday in the basement of the Federal City Shelter, which is operated by the Community for Creative Non-Violence at Second and D streets NW in a gentrifying area, homeless advocates said that their complicated legal agreement between the federal government and the city expires in three years. The shelter houses about 1,300 single men and women.

Although the city has created a task force to look at the shelter, long troubled with safety, maintenance and other issues, homeless advocates look at the cranes and new buildings surrounding them and fear the worst.

“What we are trying to prevent is for the city to ignore us until July 2016, and they say, ‘You don’t have the legal right to shelter here. Everybody out,’ ” said Eric Sheptock, a shelter resident and homeless advocate who runs Shelter, Housing and Respectful Change (SHARC). “And then they put up high-end condos and office buildings, and the homeless are pushed out of the city.”

D.C. Council member Jim Graham (D-Ward 1), who chairs the CCNV Task Force, said the group has homeless people, not condos, uppermost in mind. “The task force is going to look at how to take steps to make it a better place for homeless people.”

The task force will meet Thursday to begin discussing options for the site.