Three months after D.C. entered the second phase of coronavirus restrictions, the city’s top public health official introduced new criteria Monday for determining when to lift additional restrictions and enter Phase 3.

The list of 10 criteria includes some goals the city already has met, including a coronavirus test positivity rate below 3 percent and the ability to contact almost every new patient within a day of testing positive. D.C. is far from reaching other goals, especially a requirement that more than 60 percent of new cases be closely connected to other known cases — a metric that now stands at 6 percent.

D.C. leaders say a move of the next reopening phase would allow public schools to reopen and additional capacity for activities already permitted.

The new list eliminates a metric that had long been one of the city’s most prominent and questionable measures of coronavirus progress: community spread. D.C. relied on that count — which considered the number of cases outside of settings such as nursing homes and the D.C. jail — to determine when to enter the first two phases of reopening.

But the metric, used by few other jurisdictions, was criticized by some, and on Monday, LaQuandra Nesbitt, the city’s top public health official, said D.C. will no longer require declining community spread to move to Phase 3.

New metrics include a requirement that, on average, fewer than five new daily cases per 100,000 residents be identified (it’s now 7.4 cases), that test results come back within two days (it’s now 2.1 days) and that more than 80 percent of people who test positive complete a contact tracing interview within three days (two-thirds are now participating in interviews, although the city is calling nearly every patient).

Nesbitt also lowered the bar for utilization of the city’s hospital beds.

The Phase 2 requirement was that fewer than 80 percent of hospital beds be occupied. On Monday, with 83 percent of city hospital beds full, she put the bar at 90 percent. She said previous months have shown hospitals are prepared to accommodate a possible surge in coronavirus patients.

Nesbitt said more changes are possible as public health knowledge about the virus grows.

“I would not commit to never changing the threshold of a metric,” she said.

Entering Phase 3 would require meeting all 10 criteria, Nesbitt said, although Mayor Muriel E. Bowser (D) said Monday that she is considering lifting bans on some activities during Phase 2. In the next two weeks, she said she might reopen city-owned indoor pools, allow more activities in libraries and grant waivers for more activities at some businesses, such as live music.

Nesbitt, who outlined challenges last week with the city’s contact-tracing efforts, discussed recent cases Monday that highlight difficulties in reaching residents who might have been exposed to the virus.

Nesbitt said Monday that on 139 occasions, D.C. has sent a contact-tracer to a patient’s home when they couldn’t finish a phone interview. Nesbitt said 16 people who had tested positive for the virus were not home in quarantine.

In one case, a man visiting D.C. from out of the country couldn’t quickly be reached by phone. Contact tracers eventually informed him he tested positive, and the man said he couldn’t stay in a hotel room with his immunocompromised uncle. The city found him another room, where he stayed until he returned to his home country.

In another case, a man who tested positive insisted that he had never even been tested for coronavirus, saying he would not quarantine. Nesbitt said the city is paying “special attention” to the residential building where the man lives and works as an employee to see whether others become infected.

She said the health department told the building’s management about the man’s positive test result but did not inform residents of the building about their risk to protect the patient’s privacy.

D.C. also announced Monday changes to its list of states that require visitors to quarantine for 14 days, adding nearby West Virginia and Delaware.

An order from Bowser identifies “high-risk” states as those with a seven-day moving average of new daily coronavirus cases at 10 or more per 100,000 people. The city requires anyone in the nation’s capital for nonessential activities from those states to quarantine.

Delaware has been on and off the list as its metrics fluctuate, with state leaders last month requesting that it be removed ahead of the Labor Day holiday.

Three other states — Alaska, Arizona and Wyoming — also were newly added to the list, while California, Hawaii and Ohio were removed. The list, which exempts Maryland and Virginia, is updated on Monday every two weeks.

In a small sign of a return to normalcy in D.C., the Smithsonian Institution announced Monday that the National Museum of American History and the National Museum of the American Indian will reopen Friday with safety protocols in place. The announcement comes after four Smithsonian museums reopened in the past week.

Restrictions also were lifted Monday in Maryland.

The state began allowing restaurants to serve patrons indoors at 75 percent capacity — up from 50 percent — in jurisdictions where local leaders agree it is safe for larger crowds. Gov. Larry Hogan (R), who announced the change Friday, has cited the state’s improving coronavirus metrics in making the decision.

County leaders in the harder-hit Washington region were more reluctant to move forward with expanded dining capacity. Officials in Montgomery, Prince George’s and Anne Arundel counties aren’t allowing more customers inside restaurants, although Howard County is permitting the expansion.

In Frederick County, leaders hit pause Monday on plans to expand indoor dining over concerns that daily case counts spiked over the weekend. Forty-five new cases were reported Saturday, the most in a single day in the county since mid-July.

“Given how our health metrics have suddenly changed, taking a pause is the right approach,” County Executive Jan Gardner said in a statement.

Frederick had planned to follow Hogan’s recommendation that indoor dining capacity increase, but that expansion is on hold.

The greater Washington region on Monday reported 1,062 new coronavirus infections and 11 additional deaths. Virginia had 627 new cases and six deaths; Maryland had 412 new cases and four deaths; D.C. had 23 new cases and one additional death.

The region’s caseload Monday was the lowest in a single day since July 6, although numbers are often lower on Mondays. In Maryland, health officials on Monday reported a record low seven-day test positivity rate of 2.75 percent.

Erin Cox contributed to this report.