Local health authorities in hot spots around the U.S. are seeking to blunt the highly contagious Delta variant of Covid-19 by trying to boost stagnating vaccination rates while largely stopping short of reinstituting rules and restrictions.

During previous surges, state and local leaders attempted a variety of approaches to slow infections. They instituted mask mandates, shut down public spaces and ordered businesses to close or run at limited capacity, in efforts to buy time until vaccines arrived.

Now, many leaders are focused on campaigns of persuasion and education. In some places, frustration is mounting at the slow pace of vaccinations, while infections are accelerating.

“I’m mad, I’m upset, I’m depressed because we’re going to watch people needlessly die over the next month or two for no good reason,” Thomas Dobbs, Mississippi’s state health officer, said at a briefing Tuesday.

Thirteen hospitals in the state have run out of ICU capacity, and many others have less than 10% of beds available, Dr. Dobbs said. Mississippi has one of the nation’s lowest vaccination rates, with 33.8% of the population fully vaccinated, federal data show.

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Colorado’s Mesa County has plenty of vaccines on hand, its public health department’s executive director, Jeff Kuhr, said, but only 42% of eligible residents are fully vaccinated. The county of about 154,000 has more Delta cases than anywhere else in the state, including large metropolitan areas like Denver. In an effort to boost the vaccination number, his office now has a mobile vaccine van and a team of public-health nurses who do in-home vaccinations.

The seven-day average for new U.S. Covid-19 cases has tripled from a month ago to 34,730 but remains well below the peaks reached in January, when a wintertime surge was raging, according to Johns Hopkins University data.

The Delta variant has become a prime driver behind rising cases nationwide. The highly transmissible form of Covid-19 now accounts for 83% of sequenced patient samples, Centers for Disease Control and Prevention Director Rochelle Walensky said Tuesday.

The federal government is largely focused on vaccine messaging after the U.S. missed President Biden’s goal of getting 70% of the adult population to take at least one vaccine dose by July 4. By Tuesday, that number was 68.3%, according to the Centers for Disease Control and Prevention. Beneath that figure are deep disparities, from high rates in New England to low levels in the Deep South.

Biden administration officials have emphasized the threat posed by the Delta variant to unvaccinated people and deferred to local policy makers when asked if the uptick in cases could necessitate a return to mask-wearing and other large-scale mitigation measures.

The seven-day average for new U.S. Covid-19 cases has tripled from a month ago to 34,730, according to Johns Hopkins University data.

Photo: Mario Tama/Getty Images

Public-health researchers—including former Trump administration Surgeon General Jerome Adams and Caitlin Rivers, an infectious disease epidemiologist at Johns Hopkins University’s public-health school—are leaning on the Biden administration to take more forceful action to empower states and localities to beef up their defenses.

Some of them have said the federal government should revert to its previous guidance, which called for wearing masks indoors, regardless of vaccination status.

The critics, many of whom are supportive of the Biden administration’s overall approach to the pandemic, are also pushing for faster full-approval of coronavirus vaccines. Formal approval, rather than the emergency use authorization that currently exists, could budge vaccine skeptics and make it easier for employers and the military to mandate shots, these researchers said.

“With over 300M doses given in the US, there is ample safety & efficacy evidence,” Dr. Zeke Emanuel, an oncologist who served on President Biden’s Covid-19 transition team, tweeted on July 6. “Moving from emergency authorization to full approval will help increase vaccine uptake.”

The White House didn’t respond to a request for comment.

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“Our biggest concern is that we are going to continue to see preventable cases, hospitalizations and, sadly, deaths among the unvaccinated,” Dr. Walensky said at a press briefing Friday.

Last week, Los Angeles County became the first highly populated local jurisdiction to re-implement an indoor mask requirement, driven by a sharp rise in Covid-19 infections there. The local sheriff has said he won’t enforce the new rules, however, which leaves them largely in the hands of businesses.

Hilda L. Solis, chairwoman of the Los Angeles County Board of Supervisors, said in a statement that the new mandate was driven by increases in both the positivity rate and hospitalizations. Still, with most of the new cases concentrated among unvaccinated residents, Ms. Solis stressed that getting more people immunized would ultimately be the key to blunting the Delta variant.

In other places, health authorities are asking people to cover their faces again to stop the spread but not making it mandatory.

Though Massachusetts has one of the nation’s highest vaccination rates, the town of Provincetown, at the tip of Cape Cod, advised people on Monday to wear masks indoors when social-distancing isn’t possible. The town cited rising cases after the July 4 holiday among vaccinated and unvaccinated people.

“When you’re dealing with close quarters and indoors, and you don’t know or have the ability to determine if people are vaccinated or not, masks will continue to be an important part of keeping people safe,” Republican Gov. Charlie Baker said during a press conference last week.

New Covid-19-related rules are a tough sell in many parts of the country, including Arkansas, where Republican Gov. Asa Hutchinson signed legislation this spring barring local-level mask mandates. Covid-19 infections have been surging there, with the seven-day average for new cases topping 1,000 in recent days, and the state has one of the nation’s lowest vaccination rates.

“It’s a grave concern,” Little Rock Mayor Frank Scott Jr. said in an interview. He said he doesn’t want to require people to do things but that the city is exploring whether there may be a loophole in the new law while largely focusing on getting more shots into arms.

“You don’t want to have mandates, but if there’s a need to keep people alive, that’s something you want to consider,” the mayor, a Democrat whose office is nonpartisan, said.

Mr. Hutchinson this month began hosting a listening tour around the state aimed at encouraging people to get vaccinated. In a statement, the governor said the state is also boosting its marketing campaign to reach populations with lagging vaccination rates while marshaling a private-physician network for outreach.

“It’s not about what the government tells people to do,” Mr. Hutchinson said. “Arkansans have the information to make good decisions.”

In Pima County, Ariz., which includes Tucson, public health director Theresa Cullen said she was hopeful that the county could stymie the Delta variant with a focus on areas and demographics where vaccination is lagging.

In May of last year, Arizona Gov. Doug Ducey barred local municipalities from issuing their own Covid-19 restrictions. While Dr. Cullen said she could still issue a strongly worded recommendation on mask wearing, she didn’t yet think it was necessary in her area because the new cases hadn’t accelerated enough yet.

Delta is inflating cases in Mobile County, Ala., where health authorities say there were 62 people hospitalized with Covid-19 on Sunday, up from 15 about a month earlier. About one-third of Alabama’s population is fully vaccinated, CDC data show, the lowest level in the nation.

Republican Gov. Kay Ivey last week spoke out against any coronavirus-related mandates. Speaking to local media, she said the vaccines are safe and free and that people just need to show up and get inoculated.

Laura Cepeda, chief medical officer for Mobile County’s health department, said the vaccine push has met resistance from people who are skeptical of the shots and, in some cases, the disease itself. Nevertheless, health authorities continue to explain the vaccines and the facts, including in a Facebook chat Monday.

She said any shutdown of businesses or schools would likely not be triggered by government action but by people being out sick.

“I’m not saying we’re not going to see that, but it’s going to be driven by Covid itself,” Dr. Cepeda said.

—Brianna Abbott, Stephanie Armour and Sabrina Siddiqui contributed to this article.

Write to Jon Kamp at jon.kamp@wsj.com and Dan Frosch at dan.frosch@wsj.com