Eleven states and the District of Columbia have opened enrollment under the Affordable Care Act to allow laid-off workers to get subsidized health insurance, and the Trump administration, which has been gunning to repeal the law, is considering opening the federal exchange to new customers.

The new enrollment periods will ease insurance sign-ups for people who have recently lost health coverage along with their jobs. And they will provide an opportunity for people who didn’t buy insurance for the year to reconsider that choice.

California, Colorado, Connecticut, Maryland, Massachusetts, Minnesota, Nevada, New York, Rhode Island, Vermont and Washington are the states that run their own exchanges that have made the move to open enrollment.

In California, state officials had already extended the normal enrollment window because of changes in state policy. But Peter Lee, the executive director of that state’s marketplace, Covered California, said he was doing everything he could to alert residents that they could buy insurance if they needed it now.

“There’s no economic or public health rationale to not open the doors wide in the face of the pandemic,” he said.

Rhode Island officials also chose to create a special enrollment period to prevent people from hesitating to get testing or treatment if they were uninsured, said Lindsay Lang, the director of the state’s marketplace, HealthSource RI.

“We don’t know how big this will be in Rhode Island,” she said. The hope, she said, is to protect individuals and the state’s health system from large sums of uncompensated care as they wait for federal officials to take action, adding, “This is what we can do while we wait.”

The Affordable Care Act was set up with a short annual window for insurance sign-ups. That was meant to encourage Americans to buy coverage while they were still healthy, keeping down premiums. Under the Affordable Care Act, people who lose insurance coverage when they lose their job are already allowed to buy their own insurance. But the creation of a broad special enrollment period would mean that anyone without comprehensive insurance could simply sign up for a health plan, without having to prove such special conditions. Early reports from states suggest that a recent wave of layoffs will leave millions of Americans in need of new health coverage.

The federal government has established this type of special enrollment period in the past, typically during natural disasters. But over all, the Trump administration has worked to shorten enrollment periods and tighten eligibility rules for those seeking exceptions.

It is not clear whether the administration will establish a special enrollment period for the 32 states with markets it manages. (There are also six state-run exchanges that use the federal platform, and those states can’t proceed without federal sign-off.) The administration continues to argue in court for the law’s invalidation. In a press briefing on Sunday, President Trump reiterated that he would like to eliminate Obamacare altogether and replace it with an unspecified program he prefers.

“What we want to do is get rid of the bad health care and put in a great health care,” he said, in response to a question about the lawsuit.

There are also an estimated 17 million people already uninsured but eligible for marketplace coverage, according to a recent study from the Kaiser Family Foundation. That study found that more than a quarter of those people were eligible for a bronze plan that would cost them nothing in premiums after federal subsidies were applied (they would still have a high deductible). A broad special enrollment period could protect that group from big bills, too, if they contract the disease known as Covid-19.

“If open enrollment were more broad, and there were fewer barriers, that could make it easier for people to sign up,” said Cynthia Cox, a vice president at Kaiser and a co-author of the study.

People in so-called short-term, limited-duration health plans — those offering skimpier coverage that doesn’t meet all the requirements of an A.C.A. plan — could also sign up. Although the administration has encouraged the availability of alternatives, many may now want more comprehensive coverage.

Washington State, which has been enrolling people since March 10, has had 2,9730 residents indicate they plan to sign up as of last Thursday. About 500 have actually done so. In New York, during the first four days of the enrollment period, 150 people signed up, according to state officials.

In Rhode Island, which has had open enrollment since March 14, “we’ve had a really strong response,” Ms. Lang said. As of Friday, 233 people had enrolled, with a further 150 or so in the process of doing so.

For Americans whose income has dipped low enough to qualify them for Medicaid, that program accepts applications all year long. In the 36 states that expanded their Medicaid programs under the Affordable Care Act, that means anyone now earning less than 138 percent of the federal poverty level — about $17,000 for a single person and $35,500 for a family of four, annually — can qualify for coverage right away. Eligibility rules vary in the other states. Many people who have lost their jobs and have very low incomes are now likely to qualify.

Marketplace coverage is more complicated: In addition to the need for an enrollment period, enrollees typically qualify for financial assistance with their premiums based on their income declared on their last tax return. Individuals can use a different estimated annual income but may have to provide documentation that their circumstances have changed, Ms. Cox said.

Anyone who already has marketplace coverage but has had an income change can return to the marketplace to apply for an increased subsidy. This is true even in states that have not yet established a special enrollment period.

Initially hesitant to reopen the federal marketplaces, health insurers recently began pushing for a special enrollment period to insure people who suddenly find themselves without a job. Last Thursday, the two main trade associations sent a letter to Congress calling for a special enrollment period, allowing people to sign up regardless of whether they’re currently insured.

While it is true that people can already apply for coverage when they lose their employer-based coverage, a special enrollment period would require less paperwork and could be the quickest way for people to get coverage, said Justine Handelman, a senior vice president for the Blue Cross Blue Shield Association.