Then on Tuesday, in a bid to ensure coverage for all who want it, the administration provided details of the “special enrollment period” for some people — but not all — who miss the deadline.

“If you weren’t able to enroll in an insurance plan by Dec. 23 because of problems you had using HealthCare.gov, you still may be able to get coverage that starts Jan. 1,” the administration told visitors to the website in a message posted on the health insurance blog at HealthCare.gov. The message also highlighted the assistance available to shoppers.

“Couldn’t enroll by December 23?” it said. “We can still help you get covered.”

“Even though we have passed the December 23 enrollment deadline for coverage starting January 1, we don’t want you to miss out if you’ve been trying to enroll,” the administration said. “Sometimes despite your best efforts, you might have run into delays caused by heavy traffic to HealthCare.gov, maintenance periods or other issues with our systems that prevented you from finishing the process on time. If this happened to you, don’t worry — we still may be able to help you get covered as soon as January 1.”

If consumers want to plead their case, the administration said, they can call a special number (800-318-2596) and explain why they could not complete their applications in time.

“Tell our customer service representative that you’ve been trying to enroll and explain why you couldn’t finish by the deadline,” the website advises consumers. “They can tell you what you can do to finish your enrollment and still get covered for 2014.”

The administration did not set a deadline for such requests. The call center is normally open around the clock but is closed on Christmas, officials said.

Ms. McGuinness, the White House spokeswoman, said there were “10,000 call center workers working tonight on Christmas Eve to get people health insurance before the final deadline.” Julie Bataille, a spokeswoman at the federal Centers for Medicare and Medicaid Services, said people who had problems with the website before Tuesday could receive “individual assistance” if they contacted the call center.