WASHINGTON — Although the health care exchange enrollment deadline was March 31, the Obama administration, citing a last-minute surge in demand, built in some leeway for those consumers who were actively engaged in the process of signing up for health insurance.

As of late March, more than 6 million people had signed up for medical insurance plans through federal and state exchanges.

When the health care website, HealthCare.gov, was overwhelmed by people trying to enroll in late December to gain health coverage by January 1, the administration enacted a similar stretch of the deadline for those who tried to enroll but did not make the deadline.

In late March officials said they would loosen the deadline for those with “complex or extenuating” circumstances that keep them from fully enrolling by March 31, the last day to buy health insurance to avoid paying a fine with 2014 taxes. Several states operating their own exchanges took similar steps as the deadline ­approached.

Under the move planned by the administration, some people were to be given a special enrollment period, beyond the deadline, if they could show they were not able to enroll because of an error by the federal exchange or by the Department of Health and Human Services. The extent of the grace period was unclear at presstime.

Preceding the March 31 deadline, the Covered California exchange reached a deal with CVS Caremark Corp. and the Ralphs grocery chain to promote the health care law inside their stores. An exchange spokesman said the retailers were providing the display space at no cost to the state.

CVS has been promoting the Affordable Care Act at its stores since enrollment began in October. In California, the exchange said CVS would provide additional information in 120 stores in areas with the highest concentration of residents eligible for federal premium subsidies.

Insurers were using a number of unconventional strategies to encourage people to sign up. In Philadelphia, for example, Independence Blue Cross held an “enroll-a-thon” that included a 16-hour marketing blitz on a local television station. In Mississippi, Humana Inc. launched a bus excursion making stops at churches and community centers where it solicited last-minute signees.

The Obama administration turned to some unusual strategies as well. The president’s outreach included meeting with YouTube personalities and conveying his message over daytime television shows.