WASHINGTON - Obama administration officials said that, even after Tuesday's extended deadline, they would try to arrange coverage starting Jan. 1 for people who have had trouble getting through the sometimes-balky enrollment website.

"Our highest priority is making sure that everyone who wants to enroll to have health care coverage by Jan. 1 is able to do so, particularly since consumers had a hard time accessing HealthCare.gov in October and November," administration spokeswoman Julie Bataille said in a statement.

"As such, we are making sure we can provide information directly to consumers if and when they have questions about their particular situation, and if they are covered as of Jan. 1.

"Consumers who tried to enroll prior to today and had problems with the system should contact the Marketplace call center for individual assistance" starting Thursday, the statement said. Administration officials said they would work with insurance companies in an effort to keep people from going without coverage.

The stretching of the deadline directly affects consumers in 36 states who use the federal website to sign up for health coverage under the new health care law. Deadlines vary in the 14 states and the District of Columbia, which use their own websites.

The announcement was the latest move by the administration to deal with problems caused by the collapse of the website in October. The original deadline for consumers to sign up for coverage that would take effect Jan. 1 was Dec. 15. Last month, the White House extended that deadline to Dec. 23. On Monday, that was extended until Tuesday.

How many people have signed up for coverage under the new law remains undetermined. President Obama said Friday that roughly one million Americans had signed up using either the federal or state websites. But the sites have seen heavy traffic over the weekend and through Tuesday.

Officials said that HealthCare.gov had about two million site visits Monday and that an additional 250,000 people contacted the system's call centers. Because of heavy volume, 129,000 people were placed on the system's waiting queue, in which people would be contacted by e-mail when the website was less busy.

Several states have reported notable increases in enrollments in the last several days. In Colorado, for example, more than 5,000 people signed up Monday, bringing that state's total to 43,000, officials said. New York reported an additional 25,000 people enrolled Monday, and California has also reported late surges in enrollment.

Republicans have criticized the administration's repeated deadline extensions, saying the administration is stretching the law to make enrollment numbers in the new year look at large as possible.

"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.

Obama has said the changes are simply "common-sense" efforts to make sure that people aren't frozen out of coverage.

The political squabble has limited practical impact, however, since this month's deadline is far from the last opportunity to sign up. Instead, it's just the last chance to sign up for coverage that will take effect Jan. 1, which is primarily important to people with health problems who can't afford a gap in coverage.

Open enrollment for health benefits under the law extends through the end of March. Insurance industry officials have said most health plans will accept coverage through Jan. 10.