“Any attempt to get support or services for them is stymied because you don’t know where they’re going to end up,” said Lynn M. Kelly, the director of the Justice Center.

Image The little-known detention center in Greenwich Village, on the fourth floor, reopened last year. Credit... Suzanne DeChillo/The New York Times

When she asked that the lawyers’ letters of legal advice be forwarded to detainees who had been transferred from Varick, she said the warden balked, saying he had to consider the financial interests of his private shareholders: 1,200 members of a central Alaskan tribe whose dividends are linked to Varick’s profits under a $79 million, three-year federal contract.

Federal officials would not discuss their transfer policies, but asked for patience as they try to make the detention system more humane and cost-effective.

“We inherited an inadequate detention system from the previous administration that does not meet ICE’s current priorities or needs,” said Matthew Chandler, a Homeland Security spokesman. Officials say they are committed to a complete overhaul, including less-penal detention centers with better access to lawyers.

The volunteer lawyers and the petition’s author, an ailing refugee from torture in Romania who spent eight months inside Varick, say many problems persist there, though the added scrutiny has led to improvements. Detainees who want a Gideon Bible no longer have to pay the commissary $7. Immigration officials are more responsive when a lawyer complains that a detainee in pain is not getting treatment.

But most detainees do not have a lawyer, and the few who do include men who have fallen prey to incompetent or fraudulent practitioners. Recurrent complaints include frigid temperatures, mildew and meals that leave detainees hungry and willing to clean for $1 a day to pay for commissary food. That wage is specified in the contract with the Alaskan company, which budgeted 23,000 days of such work the first year, and collects a daily rate of $227.68 for each detainee.

The Alaska connection is one of the stranger twists in the jail’s fitful history. Opened as a federal immigration detention center in 1984, Varick became chronically overcrowded after 1998, when new laws mandated the detention of all noncitizens who had ever committed a crime on a list of deportable offenses, expanded to include misdemeanors like drug possession.