Connecticut recorded 41 fewer coronavirus-related hospitalizations Saturday, continuing the downward trend it saw over the last week.

But even as officials noted that certain numbers looked encouraging and anticipated the partial reopening that Gov. Ned Lamont has set for May 20, some municipal leaders have taken a cautious tone about moving forward.

Fatalities continue to climb, reaching 2,436 — another 97 deaths since Friday, according to data from the state Department of Public Health, which indicated a total of 29,287 residents have tested positive for COVID-19 since the crisis began — up 523 from Friday.

As sunny weather arrived and more state residents headed outdoors — Sleeping Giant State Park in Hamden was closed by 10:30 a.m. Saturday because it had reached capacity — both state and local officials emphasized the importance of continued social distancing.

Katie Dykes, commissioner of the Connecticut Department of Energy & Environmental Protection, held a virtual press conference from Gillette Castle State Park “reminding residents to continue to practice safe social distancing when visiting state parks, especially as more consistent nice weather arrives,” according to a DEEP tweet.

Dykes’ Saturday conference was just the latest in the department’s attempts to spread the message about social distancing.

“When a park has closed to capacity, do not park elsewhere and walk in,” the DEEP tweeted earlier in the day, also asking visitors to leave space on trails so that others can pass.

On Friday, the department asked park visitors to remain 6 feet apart from others and to carry masks, according to a release that also recommended exploring less-popular parks to avoid crowds.

“If you do visit a state park or forest, it’s important to do so safely,” the release said. “Parks remain open for solitary recreation or recreating with members of your immediate household in groups no larger than five people. Picnics are not permitted at this time, and DEEP recommends that you keep your visits to state parks brief.”

Municipal leaders shared their own messages about social distancing.

“It is vital that you keep doing what you have been doing to prevent the spread of the virus — practicing social distancing, wearing cloth face coverings, and avoiding all gatherings,” New Haven Mayor Justin Elicker said in an email update Friday. “We are not out of the woods. In fact, we have a long way to go and must be very careful and deliberate as we consider opening up.”

Meanwhile, in Danbury, Mayor Mark Boughton took to social media, urging folks to “follow the guidance.” The city had lost another resident to the virus, he noted.

Bridgeport Mayor Joe Ganim again asked residents “to please continue to stay safe and stay home” in a video posted to his Twitter account.

And in the small town of North Branford, whose officials used Facebook to hold a press conference Friday, police chief Kevin Halloran suggested that recent rainy weather may have kept violations of social distancing guidelines at bay, noting that the town had not seen many issues with large gatherings.

While local leaders said data indicated the rate of increase of COVID-19 cases in town was slowing, they repeated the same message as the DEEP chief and city mayors: Don’t stop social distancing.

“As we’re seeing a flattening of the curve ... this not time for a victory lap,” said James Buck, North Branford’s emergency management director. “Stay safe. Stay home.”