"You should not wait until tomorrow. If you are aiming to get coverage Jan. 1, you should try to sign up today," said Julie Bataille, a spokeswoman for the federal agency overseeing the overhaul.
Bataille said the grace period was being offered to accommodate people from different time zones and to deal with any technical problems that might result from a last-minute rush of applicants.
Monday had been the deadline for Americans in the 36 states served by the federal HealthCare.gov website to sign up if they wanted coverage upon the start of the new year. Some other states have also extended enrollment deadlines slightly.
The HealthCare.gov site had a disastrous, glitch-prone debut in October, but the government reported on Twitter that it was running smoothly Monday morning.
As the Monday deadline drew new, last-minute health insurance shoppers called help lines and attended enrollment events. More than 1 million people visited the refurbished website over the weekend, and a federal call center received more than 200,000 calls.
The original sign-up deadline already had been pushed back a week because of the technical problems that plagued the federal marketplace for weeks, but hundreds of upgrades to storage capacity and software have cut error rates and wait times.
"It's just nonstop now. Everybody knows about it. Everybody wants it," said Florida enrollment counselor Madeleine Siegal. She said her organization in Fort Lauderdale was slammed with walk-ins and appointments Friday, had several weekend enrollment events and opened its doors an hour early on Monday.
Ronald Bellingeri, a 59-year-old general contractor in Florida, signed up Friday in 90 minutes with help from an enrollment counselor. Bellingeri said he waited until the last minute because he didn't know what to do or where to go.
On Friday, he chose a gold plan with a $156 monthly premium. The government is picking up $472 per month because of his income.
"I just walk in the door and an hour and a half later, I have health insurance. It makes me feel great," he said.
Roger Colyn, 60, of Des Moines, was happy when he left his Monday morning appointment with an Iowa enrollment navigator. She helped him sign up for a silver-level health insurance plan, that will cost him $10.79 in monthly premiums after tax credits and government aid are factored in.
"I feel relieved," said Colyn, who had previously tried to sign up at a local hospital, but had run into problems when he was told the cost would be more than he could afford. He started the process over Monday and was told he would have coverage starting Jan. 1.
Others said they will let the date pass without making a decision.
"I'm in no hurry, though it'd be nice to be able to visit a doctor without stress," said Kyle Eichenberger, an uninsured 34-year-old from Oak Park, Ill.
Eichenberger said he hit a wall on the website when he first tried to enroll. More recently, the 34-year-old stay-at-home dad hasn't had time to get his questions answered. His family situation doesn't fit into a neat category: His children and their mother have health insurance through her job. But her employer doesn't offer coverage to opposite-sex domestic partners, and the couple are not married.
"I'm an Obamacare supporter, though I think it is full of problems," Eichenberger said. "It's better than the system we had before and already helps my kids get free preventive care. I'd like to see the whole system streamlined to be more user-friendly. Keep the basic idea, but don't make me feel like I'm navigating a maze to get a simple checkup."
Minnesota, one of the states running their own insurance exchanges, had long planned a Monday deadline to sign up for coverage starting Jan. 1. But amid problems with its website and extra-long hold times to reach its help center, the state last week postponed the deadline by eight days, letting people sign up through Dec. 31.
Maryland has also extended its deadline, to Dec. 27.
In Washington state, people who got up early on Monday to start their application for health insurance on the state's online marketplace may have been disappointed. The site was down for maintenance until nearly 9 a.m.
Associated Press writers Catherine Lucey in Des Moines, Kelli Kennedy in Fort Lauderdale, Fla., Patrick Condon in St. Paul, Minn., Brian Witte in Annapolis, Md., and Donna Blankinship in Seattle contributed to this report.
Associated Press Medical Writer Carla K. Johnson can be reached at http://www.twitter.com/CarlaKJohnson