New Jersey approved outdoor graduations and indoor training for professional sports teams on Tuesday, while Delaware plans to lift its ban on short-term rentals, eliminate its quarantine on out-of-state residents, and allow outdoor gatherings, including graduations, of up to 250 people starting Monday.

But even as the Philadelphia region prepared to move to the “yellow” phase of reopening on June 5, Pennsylvania officials were more cautious. Gov. Tom Wolf said Tuesday he had concerns about large gatherings even in the least-restrictive green phase and did not say whether he would approve outdoor graduation ceremonies.

And Philadelphia Mayor Jim Kenney warned that plans to reopen could be derailed by people attending large gatherings and flouting safety recommendations, citing a Memorial Day party in Brewerytown where police broke up a gathering of 200-plus people listening to a DJ.

“There’s nothing that I would want more than to go to yellow and to go to green on schedule,” Kenney said, “but the frustration about this is that when you see 250 people at 28th and Cecil B. Moore, or 2,000 people in a pool in St. Louis by a lake without masks on, the possibility of us getting where we need to be or where we want to be in a timely manner gets continually diminished.”

For now, signs continued to be good: Philadelphia on Tuesday reported fewer than 100 new confirmed coronavirus cases for the first time in weeks, Public Health Commissioner Thomas Farley said.

A month ago, the city regularly reported 500 or more cases per day; the 97 new positive cases confirmed Tuesday are the latest sign that the virus’ spread in the city is slowing.

A decline in cases was continuing statewide, as well, said Health Secretary Rachel Levine.

The state has increased its testing capacity from 50,000 to 80,000 tests per week and has increased its contact tracing from 433 coronavirus patients to 1,700, Wolf said.

They stood by their Friday decision to move Southeastern Pennsylvania to the yellow reopening phase next week even though the counties have not reached one of the commonwealth’s initial benchmarks for lifting shutdown restrictions: having about 50 new cases per 100,000 residents over 14 days.

“Over the last week, we have significantly been able to increase our testing and as we increase our testing, we’re going to be picking up more and more people with COVID-19” in case counts, Levine said. “That makes an incidence rate, such as 50 per 100,000, less useful to us as a measure of what counties can go from red to yellow and then yellow to green.”

She and Wolf also cited the expanded testing and contact tracing and residents’ dedication to social distancing and masking as reasons the area can reopen.

Montgomery County Commissioners Chair Val Arkoosh, who is also a physician, said she agreed with Wolf’s plan to move the area to yellow, saying the county’s hospitalizations continue to drop and contact tracing will soon be conducted on every new case.

Philadelphia should be able to move to yellow on June 5 as long as the “numbers continue to be good,” Farley said, though the city could choose to adopt more restrictions than the state if officials deem it necessary.

In one “step toward a new normal,” Kenney and Farley on Tuesday approved walk-up ordering at restaurants, including food trucks, which had been shut down. Their order also takes steps to implement the previously announced restart of the real estate industry and expand the construction activity allowed.

The city also broadened its testing rules, now also recommending that asymptomatic people who have been exposed to a known carrier of the coronavirus be tested.

Pennsylvania on Tuesday reported 451 additional confirmed cases for a total of 68,637, and 13 new deaths for a total of 5,152. Reporting is sometimes delayed after a weekend.

As neighboring states moved to approve them, Wolf did not say he would prohibit graduation ceremonies but said he and other state officials “think there needs to be a limit on the number of people who gather.”

The state is also still working with professional and amateur sports teams “to try to figure out what the guidelines ought to be, again to keep players and to keep citizens safe,” he said.

New Jersey reported 703 new cases for a total of 155,764 and 54 deaths — likely undercounted due to the holiday weekend — for a total of 11,191.

Gov. Phil Murphy on Tuesday said the spread of the coronavirus in New Jersey is “slowing considerably,” and noted the state is “moving closer” to its next phase of economic recovery, in which businesses such as restaurants, libraries, and museums may be able to reopen with restrictions.

“The progression across the past two weeks has been constant and undeniable,” Murphy said. “We’re seeing many more good days than bad.”

High schools and colleges will be able to hold socially distanced outdoor graduation ceremonies starting July 6, he announced. Full guidance will come Wednesday, but schools may need to hold multiple graduation ceremonies if there remains a cap on outdoor gatherings. They also must determine the “minimum number” of faculty and staff needed to run the ceremonies and coordinate with local law enforcement and health officials, and mail caps, gowns, and diplomas to students when possible, according to a preview of ceremony guidelines released Tuesday.

The governor also announced that professional sports teams may resume indoor training and practice at New Jersey facilities if sports leagues permit. The 76ers and Flyers operate practice facilities in South Jersey.

The poor weather over Memorial Day weekend deprived local officials of the opportunity to practice capacity management at Jersey Shore beaches or to learn much about what will be needed to enforce coronavirus preventative measures during a busy summer season, Murphy said, though adding he saw good compliance with social distancing during a visit to Seaside Heights.

In Delaware, residents following safety guidelines over the weekend at the state’s beaches “really impressed” Gov. John Carney.

“It shows that attitudes have changed in a significant way,” Carney said, also praising businesses. “I feel much more comfortable about moving forward because business owners are really attentive to the changes they need to make. ... I think we’re ready.”

Effective June 1, when the state moves into its “phase one” of reopening and lifts it stay-at-home order, short-term rentals will be permitted and shopping malls, retail stores, restaurants, and other businesses will be allowed to reopen with low capacity and safety restrictions, Carney announced.

Outdoor gatherings of up to 250 people — including weddings and graduation ceremonies — will be permitted with precautions.

The state’s 14-day quarantine requirements for out-of-state visitors will also lift Monday.

“We are where we are because Delawareans listened and stayed home,” Carney said in a statement. “It’s critical that Delawareans not rush out and undo all the hard work they’ve done to get us to this point. Let’s continue to be cautious and responsible as we ease our way into this new normal.”

Contributing to this article were staff writers Rob Tornoe, Sean Collins Walsh, Allison Steele, and Michaelle Bond.