Four weeks after the first COVID-19 testing site available to all state residents opened in Paramus, worried people are continuing to line up in their cars in the early morning hours waiting to be swabbed, a sign of the continuing gap between the demand for tests and the available supply.

On Thursday, they sat in parked cars for hours, queued up on Paramus Road — some sound asleep and wrapped in blankets — waiting to get tested in the parking lot of Bergen Community College.

“I’ve been here since 3 o’clock,” said Ken Roy of Hackensack. “We need the test. We have no choice.”

When it debuted March 20, the first of two federally-supplied sites open to all state residents without an appointment, lines formed early and went for miles. Some who arrived late often got turned away when tests ran out.

State officials say there are now 60-odd testing sites in New Jersey, but most are either private and require an appointment or restricted to residents of the cities or counties that sponsor them.

“Somewhere in Newark they have appointments,” said Abraham Amankwah of Irvington who had been on line since 3:30 a.m. “I called yesterday. They told me they were already booked up with appointments, so I should come down here.”

Those on line are not the only people in New Jersey frustrated by ongoing shortages of testing capability.

Gov. Phil Murphy has repeatedly talked about the need to greatly expand testing, both to get a better handle on the current health crisis and to prepare for the key role it could play in safely emerging from the shutdown.

“We recognize that significantly expanded testing is critical and necessary for us as we move forward,” he said Thursday during his daily COVID-19 press briefing in Trenton. “I continue to press this case in my conversations with the White House and our other federal partners both on my own and alongside all of our governors.”

He said he commiserated with those looking to be tested, and noted that test kits, nose swabs and other materials remain in short supply.

“This isn’t unique to New Jersey. In fact we’re punching well above our weight,” he said. “But if you’re waiting for a test and you’re in your car and you’re frustrated, I don’t blame you. We need to get more testing materials to New Jersey.”

Murphy said he intended to address the issue of expanded testing capacity during an afternoon discussion with Vice President Mike Pence and perhaps President Donald Trump, who has repeated talked of the need to get the national economy back on its feet soon.

“We need robust health care infrastructure in place, including especially the ability to mass test and get quick turnaround,” Murphy said. Enhanced testing is the cornerstone of the plan to move forward once the acute health crisis has abated, he added.

“The reason we need to have that is so we can address … where are the hot spots,” Murphy said. “Let’s be able to identify those hot spots, have a robust contact tracing infrastructure in place, a plan to quarantine and self-isolate. That allows us … the confidence that we can begin to open back up again, without running the risk of something coming in through the back door with the equivalent of throwing gasoline on the fire and going right back through the hell again that we’re going through now.”

Murphy also said following that protocol might also buy time until the development of a specific COVID-19 treatment or ultimately a vaccine.

To be sure, the state has made significant strides in the amount of testing being done. Also at the daily briefing, Edward Lifshitz, the medical director for communicable disease service at the state Department of Health, estimated that between 7,000 and 9,000 tests were being conducted daily now, including negative results, according to reports from the major laboratory companies.

“There’s obviously more opportunity for folks to be tested, perhaps closer to home, at a local county site or even at their individual health care provider,” said Assistant Health Commissioner Chris Neuwirth, of the 66 sites in the state, including city, county and private facilities.

Meanwhile, the private labs say there’s no longer a massive backlog of tests waiting to be processed. In fact, they say, they now have excess capacity.

“The commercial labs, the state lab, and others have learned a lot over the past few weeks about how to increase their capacity to test,” Neuwirth said.

Nearly 140,000 tests in New Jersey have been handled by the major laboratories, and 44.8% have been positive. In all, including all tests reported to the state, more than 75,000 New Jerseyans have tested positive for COVID-19.

Officials caution, though, that because tests are generally only being conducted on those reporting symptoms, a much lower positivity rate would likely be reported if testing were more widely available.

Among those on line in Paramus early Thursday was Erica Sanchez who drove up from Newark at 2 a.m.

“Since we heard that they were doing testing here, we decided to come,” she said. “They told us we had to be early here in order to get tested.”

The frustrations were evident among some of those on line.

“It’s horrible,” said Carlos Poupart of Passaic. “Cause you have to leave your house about 2, just to be here. Anybody comes here, when they post at 8 o’clock, they’re going to be turned away. Cause the line’s already all the way down there, miles back.”

A county sheriff’s officer on duty at the site said the first car got in line at about midnight. On other days, he’s seen them line up as early as 10:30 the night before.

Police beeped people awake shortly before 8 a.m. and the line started to move. A total of 500 tests were available Thursday, and officers put a “250” sign at the midway point.

Meanwhile, something different did happen for the first time in Paramus on Thursday. The line got shorter and then disappeared. Murphy has also said that the FEMA test site at the PNC Arts Center in Holmdel routinely has tests left over, and has advised those looking to be tested to driving there instead.

Murphy called it a “glimmer of hope,” and Neuwirth thinks pent-up demand at these two federal sites might finally be easing.

In addition, tests at the FEMA sites are now self-administered, meaning a health care volunteer hands the test kit on the end of a pole to the car’s occupant and then instructs them on how to properly swab their own nose before getting the specimen back. That means less exposure for the staff.

Also, Wednesday marked the debut in Middlesex of what officials have called a potential game changer for testing, a saliva test developed by a team including Rutgers University scientists that holds the potential to allow more tests to be conducted more quickly and with a quicker turnaround for results. It too is self-administered, allowing those being tested to simply spit into a tube.

RELATED: New Rutgers saliva test for COVID-19 makes debut in Middlesex County

The FEMA sites will stay open through the end of May. They are operating on a rotating schedule that can be found online, along with other testing locations and options.