CHICAGO (AP) — The government’s revamped health care website was put to its biggest test yet as a record-breaking surge of Americans rushed to beat Tuesday’s extended deadline to sign up for coverage.

After a disastrous, glitch-plagued rollout in October, HealthCare.gov, the website where people in 36 states can shop for insurance, received nearly 2 million visits Monday and handled the traffic well, the government said.

Monday was the sign-up deadline for people wanting coverage at the start of the new year. But the Obama administration pushed back the deadline a day to deal with expected heavy traffic from procrastinators.

The grace period was the latest in a series of delays that have marked President Barack Obama’s health care overhaul.

Critics of the law seized on the extension as more evidence that the program is in trouble.

“The amazing, ever-expanding deadline? It’s clearly a sign of desperation by the administration to do everything they can to increase the number of people signing up,” said health economist Gail Wilensky, who ran Medicare for President George H.W. Bush.

The law’s supporters said the extra day means the public got the message and wants subsidized health insurance.

“A lot of people who previously found health care unaffordable are learning they can get very substantial subsidies that bring premiums within their reach,” said Ron Pollack, president of Families USA, a liberal advocacy group leading efforts to get uninsured people signed up for coverage next year. “That’s why we’re seeing a large influx of people trying to get enrolled.”

The website went through extensive hardware and software upgrades to make it more reliable and increase its capacity.

When the number of simultaneous users reached 60,000 on Monday, site operators employed a queuing system that allows people to either wait or give an email address to be invited back later, the government said. More than 60,000 users gave their email.

Many states operate their own online marketplaces for buying coverage, and some of them also extended their deadlines.

The insurance industry, too, has pushed back deadlines for payment, with most health plans allowing customers to pay by Jan. 10 and still get retroactive coverage to Jan. 1.

“With deadlines that keep changing, insurers want to alleviate confusion,” said Robert Zirkelbach, spokesman for America’s Health Insurance Plans. “Health plans are going to do everything they can to help consumers with the enrollment process.”

Obama said Friday that more than 1 million Americans had enrolled for coverage since Oct. 1. The administration’s estimates call for 3.3 million to sign up by Dec. 31, and the target is 7 million by the end of March. After that, people who fail to buy coverage can face tax penalties.