WASHINGTON — The Biden administration’s new website allowing people to order up to four free at-home coronavirus tests quietly went live on Tuesday — a day in advance of its formal launch — and demand already appeared to be significant.

A combined total of more than 1 million visitors were on the home page and the ordering page of covidtests.gov at one point Tuesday evening — more than 40 times as many as were on the government site with the next highest traffic, the U.S. Postal Service’s package-tracking page, according to analytics.usa.gov, which monitors traffic on participating federal websites.

At a White House news conference on Tuesday, President Biden’s press secretary, Jen Psaki, said the official launch would take place on Wednesday morning, but that the site had begun taking orders during what she described as a “beta testing phase” conducted by the U.S. Digital Service, the government’s technology support division.

Also on Wednesday, the White House will announce that it is making 400 million “high quality,” nonsurgical N95 masks available, free of charge, at community health centers and retail pharmacies across the nation. The masks will be released from the Strategic National Stockpile, officials said, calling the effort the “largest deployment of personal protective equipment in U.S. history.”

The two moves show that the Biden administration is trying to step up its coronavirus response as the highly infectious Omicron variant drives a spike in cases across the nation. The administration at first resisted the idea of sending tests to Americans’ homes, but it has now enlisted the U.S. Postal Service to handle orders and deliveries.

But it wasn’t long before the pilot testing revealed some apparent glitches.

Some people living in apartment buildings said they were blocked from ordering tests if other tenants had already put in requests for the same building. Some who receive their mail at post office boxes also reported confusion in ordering, because the site includes a disclaimer that says orders would ship only to valid residential addresses.

In a statement, the Postal Service said the problems were confined to “a small percentage of orders” and recommended that customers file a service request or contact the Postal Service help desk at 1-800-ASK-USPS (1-800-275-8777).

Still, it was frustrating for those who had difficulty ordering. Gina Lindo, 46, easily ordered four tests for herself from the federal website on Tuesday afternoon, requesting that they be sent to her single-family home on Long Island. But when Ms. Lindo tried to order tests for her parents, who live in an apartment in Queens, she got an error message.

“Home Covid-19 tests have already been ordered for this address,” the message read. “We are unable to process duplicate orders for the same address.”

Ms. Lindo said she had immediately called her mother to ask if she had already ordered the tests. She had not, but an upstairs neighbor had, they learned.

“They haven’t been able to get tests, they are sold out everywhere,” Ms. Lindo said. “I know it’s probably a line in the code on the USPS website that needs to be changed, but I really do hope that they change it quickly so that we can order the tests.”

Her mother teaches English as a second language at a senior center once a week, she said, and wants to test regularly for that reason. She and Ms. Lindo’s father, who are both in their 70s, also hope to take home tests before their grandson’s upcoming 16th birthday party, and before upcoming appointments with their doctors.

Mr. Biden has experience with fumbled website rollouts. When he was vice president, he and President Barack Obama presided over the disastrous launch of healthcare.gov, the online health-insurance marketplace created by the Affordable Care Act. Jeff Zients, who is now Mr. Biden’s coronavirus response coordinator, was brought in to rescue the troubled site, which crashed repeatedly under a crush of early users. The U.S. Digital Service was created in its aftermath.

Visitors to covidtests.gov, which is also available in Spanish, can click through to a Postal Service web page where they can order four tests per household, free of charge. Orders will usually ship in seven to 12 days, the website says.

“We can’t guarantee there won’t be a bug or two,” Ms. Psaki said, “but the best tech teams across the administration and the Postal Service are working hard to make this a success.”

Matching testing supply with demand has been a persistent challenge for both the Trump and Biden administrations, and Mr. Biden has come under criticism for not ramping up the supply of rapid at-home tests quickly enough as Americans struggled to get tested amid the emergence of the Omicron variant.

Experts say there are three situations in which people should use at-home tests: if they begin to have symptoms of Covid-19; if they were exposed to someone who tested positive for the virus five or more days earlier; or if they are planning to gather indoors with someone at elevated risk from Covid-19, and want to assure themselves they are negative.

But the Omicron variant has put an intense strain on the supply of both laboratory and in-home tests. Not long after Thanksgiving, the rapid tests began quickly disappearing off pharmacy shelves, and there were long lines at clinics for polymerase chain reaction, or P.C.R., tests.

The Biden administration responded by ordering a billion at-home tests to give out free to the public, in two 500-million batches. The second batch will not be available until the spring, administration officials have said.

The administration is also making free rapid tests available at community health centers and rural health clinics. And as of Saturday, people with private insurance have been able to seek reimbursement for tests they purchase themselves. Insurers will be required to cover eight at-home tests per person per month, although some have said it may take time to put the reimbursement system into place.