A man looks over the Affordable Care Act sign-up page on the HealthCare.gov Web site in New York in this Oct. 2, 2013 photo illustration. (Mike Segar/Reuters)

Obama administration officials acknowledged Monday that they made an 11th-hour change in rules for the federal health insurance exchange to allow Americans to enroll in health plans through Christmas Eve — 24 hours later than advertised — and still get coverage that begins on New Year’s Day.

The switch occurred in the form of a software change that government officials and IT contractors inserted into the computer system over the weekend for the online insurance marketplace. Anyone who finishes enrolling by 11:59 p.m. Tuesday will have insurance on Jan. 1, the first day it becomes available.

The change is the most recent in a series of policy shifts the administration has made as a milestone approaches for what has been a tumultuous three-month start of the long-awaited opportunity for Americans to buy new health plans under a 2010 law intended to reshape the nation’s health-care system.

Administration officials did not disclose the change to the public or to most insurers participating in the new marketplace before it was reported by The Washington Post.

Under the federal rules, Dec. 23 has long been cited as the deadline for consumers to sign up for insurance if they want to have it on Jan 1. On Monday, even as they acknowledged that the rewritten computer code gives people an extra day, federal health officials insisted that the deadline had not changed, saying they were merely trying to accommodate people who might have trouble enrolling because of heavy last-minute Web traffic.

Asked about the reason for the extra day — and why it was kept secret at first — officials at the Centers for Medicare and Medicaid Services, the federal agency overseeing the health insurance exchange, at first declined to respond and then issued two statements with slightly differing explanations. Julie Bataille, director of CMS’s office of communications said in the second statement that the official deadline for signing up for Jan. 1 coverage remained Monday, but added: “[W]e have taken steps to make sure that those who tried to enroll today but had delays due to high traffic have a fail-safe.”

With the switch late and unannounced, people around the country eager for insurance treated Monday as a real deadline, swamping the federal marketplace’s Web site, HealthCare.gov, and deluging the phone lines and offices of workers trained to help consumers enroll. As the day went on, word of the brief reprieve began spreading, and consumers and their helpers expressed widespread relief.

“That gives people a little bit of time. Maybe they can do it in the middle of the night when there aren’t that many people online,” said Tiffany Green, an enrollment worker, or navigator, for Sinai Health System in Chicago, who said that she and her colleagues had managed to enroll 25 people by midday on Monday.

Through the day, HealthCare.gov had intermittent problems accepting new applications because of high traffic volume. The number of people on the Web site simultaneously peaked at about 90,000 late Monday morning, according to government figures that have not been made public. And starting at 11 a.m., consumers intermittently were placed in online queues — a relatively new feature of HealthCare.gov for periods when the site has more traffic than it can handle — rather than being allowed to apply on the spot. The system notifies people when it is safe to return, and by late afternoon, about 60,000 people had provided e-mail addresses to receive such follow-up.

But for the most part, the Web site held up, even under Monday’s duress, significantly better than during October and November, the first two months of the sign-up period, when software and hardware defects frustrated many consumers who received error messages that thwarted their ability to sign up.

The extra sign-up time means that anyone who finishes choosing a health plan in the federal exchange will have coverage as of Jan. 1, even if he or she does not begin applying until Tuesday. But it is not the last chance to get insurance through the marketplace. The open-enrollment period that began Oct. 1 runs through the end of March. In addition, an administration official, speaking on condition of anonymity about a policy that is still being finalized, said Monday that people who try to sign up for insurance by Tuesday night — but fail because the Web site does not cooperate — may appeal for special consideration.

According to two individuals familiar with the 24-hour extension, who spoke on condition of anonymity about the matter before it became public, the new software code was built into the computer system as a buffer in case HealthCare.gov faltered amid a deadline surge of customers. The additional time is automatic, they said, and cannot be overridden by individual insurance companies.

Insurance industry leaders protested the change, as they have similarly objected to other late-breaking policy shifts by the administration for the new marketplace. Robert Zirkelbach, spokesman for America’s Health Insurance Plans (AHIP), the industry’s main trade association, issued a statement Monday lamenting “potential confusion or disruption caused by all of these last minute changes to the rules and deadlines.” He said that “insurers will continue to do everything they can to help consumers through the enrollment process.”

The latest — and quietest — rule change follows a decision late last week by the Department of Health and Human Services to offer an exemption from a requirement in the health care law that most Americans have insurance as of Jan. 1. Under enforcement rules, anyone who does not select a plan by the end of March risks a federal fine. The exemption applies to people whose health plans are being canceled because the policies failed to meet new federal benefits standards.

Also last week, AHIP made a policy change on its own, saying that people who signed up by Dec. 23 would have until Jan. 10 to pay their first month’s insurance premium. People who enroll cannot actually get coverage until they pay that first bill.

During December, enrollment through the federal marketplace has increased significantly from a meager start during the first two months, but it still lags behind the administration’s expectations.

An internal HHS memo from early September, obtained by the Associated Press, projected that the enrollment nationally would hit 3.3 million by the end of December. As of Sunday, the total enrollment through the federal marketplace was roughly 890,000, according to government figures that have not been made public. That compares with about 137,000 people who signed up by the end of November, and about 227,000 more who enrolled by then through the 14 state-run marketplaces.

On Monday, the director of California’s marketplace, the nation’s largest, announced that more than 400,000 people in that state had chosen health plans as of Sunday. The state of New York reported that 137,000 people had signed up in its exchange.

Early in the day, before they became aware of the administration’s timing shift, navigators in several states said that they were inundated with walk-in clients and phone calls from people anxious for their insurance to start next month, including some who had trouble enrolling during the fall and others who ran into fresh trouble Monday morning.

The anxious clients included people with serious ailments and pending medical appointments. For example, Caitlin Zibers, a navigator at Health Partnership Clinic in Olathe, Kan., had an appointment Monday afternoon to help enroll a young man who had called in the morning saying he desperately wanted insurance because he needs a liver transplant and is scheduled to see a specialist in late January.

In Virginia, which relies on the federal exchange, walk-in traffic at the Greater Prince William Community Health Center in Woodbridge was brisk, even though the center had tried to ward off a last-minute crush by remaining open Saturday and Sunday. By 1 p.m., more than 100 people had called to ask about signing up, according to Blanca Castillo, a counselor there, who said that HealthCare.gov had been “about 50 percent working” in the previous week.

In Maryland, with its own exchange, the signup deadline is Dec. 27. The District also has a separate exchange, with different deadline for different health plans,

In parts of Kansas, navigators were having more luck. Valorie Libertus, a navigator in Emporia, Kan., said she had barely sat down at her computer at 8 a.m. when her first client of the day arrived and told her, “I hear this is where I need to be for insurance.” Within 30 minutes, the woman was enrolled in coverage and on her way to work. She said they were expecting to work after hours Monday evening and still would probably turn people away.

Told about the extra day, Libertus sounded relieved. “If we can squeeze a few more in tomorrow, I’m happy to do it,” she said. “We didn’t know, but that way we can push a few more in.”

Sandhya Somashekhar and Lena Sun contributed to this report.