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

"Our highest priority is making sure that everyone who wants to enroll to have healthcare 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.

"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," Bataille said in a statement.

"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 on a case-by-case basis in an effort to keep people from going without coverage.

Stretching the deadline affects consumers in 36 states who use the federal website to sign up for health coverage. Deadlines vary in California, 13 other 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 federal enrollment website in October. The original deadline was Dec. 15. Last month, the White House extended the deadline to Monday. On that day it was extended until Tuesday.

How many people have signed up for coverage under the new law remains undetermined. President Obama said Friday that roughly 1 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 HealthCare.gov had some 2 million site visits Monday, and an additional 250,000 people contacted the system's call centers. Because of heavy volume, 129,000 people were placed in the system's waiting queue, in which people are contacted by email when the website is less busy.

Several states have reported notable increases in enrollment 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.

In Connecticut, officials said, more than 6,700 people had enrolled Monday, bringing the state's total to 62,153. New York reported 25,531 enrollments Monday, bringing its total to 214,077.

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. President 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 effect, 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. Consumers still need to pay their initial premiums to be covered. Insurance industry officials have said most health plans will accept payment through Jan. 10.

david.lauter@latimes.com

Twitter: @DavidLauter