Vaccinated people have been burning up the phone lines at the community health center in rural Franklin, La., clamoring for the newly authorized Covid booster shot.

But only a trickle of people have been coming in for their initial doses, even though the rate of full vaccination in the area is still scarcely 39 percent.

The dichotomy illustrates one of the most frustrating problems facing public health officials at this stage of the pandemic: Almost all the eligible adults who remain unvaccinated in the United States are hard-core refusers, and the arrival of boosters is making efforts to coax them as well as those who are still hesitating even more difficult. In the September vaccine monitor survey from the Kaiser Family Foundation, 71 percent of unvaccinated respondents said the need for boosters indicated that the vaccines were not working.

“This vaccine has tested me like nothing before and I’ve been doing this for 40 years,” said Dr. Gary Wiltz, director of the Franklin health center. “I can’t tell you how many people we’ve tried to cajole into taking it.”

In some ways the Covid vaccine landscape reflects great progress: Millions of holdouts have decided to get vaccinated over the past couple months, many prodded at the last minute by mandates or anxiety over the highly transmissible Delta variant. (Three unvaccinated people who showed up for shots in Franklin the other morning came because each knew someone who had recently died from Covid.) The decline of new cases recently in many states is another marker of the success of the vaccine campaigns, public health officials say.

But millions of adults are not covered by mandates. Experts in vaccine behavior fear that the country is bumping up against the ceiling of persuadable people, one that is significantly lower than the threshold needed for broad immunity from Delta and, possibly, future variants.

“One day we just hit a wall,” said Dr. Steven Furr, who practices family medicine in rural Jackson, Ala., where he has even made house calls to give patients their Covid shots. “We had vaccinated everybody who wanted to be vaccinated and there was nobody left.”

About 56 percent of the U.S. population is fully vaccinated, a level that exceeds some early estimates about what it could take to achieve so-called herd immunity against the coronavirus. That percentage will surely rise once the shots are authorized for children under 12. But Delta is so contagious that experts have revised their optimum coverage estimates to 90 percent or higher.

According to the Kaiser Family Foundation’s vaccine surveys, those who say they will never get the vaccine — the “definitely nots” — have held steady for months between 15 and 12 percent of respondents. The rising vaccination rates of late reflect the steady shrinking of a different group — those who say they had been waiting to decide and could be convinced. They now total just 7 percent, down from 39 percent in December.

(An additional 4 percent of respondents say they would get vaccinated only if their workplace or school mandates it.)

But even as boosters are providing added protection for vulnerable populations, they are raising further doubts among people like Christopher Poe, 47, who works in a manufacturing plant in Lima, Ohio. He hasn’t gotten the shot, despite haranguing and wheedling from worried relatives. He said the need for a booster had deepened his skepticism.

“It seems like such a short time and people are already having to get boosters,” Mr. Poe said. “And the fact that they didn’t realize that earlier in the rollout shows me that there could be other questions that could be out there, like the long-term effects.”

And when shots are approved for children ages 5 through 11, as is soon expected, health officials fear that the need for boosters will make parents of those younger children, whom surveys show are very skittish about the vaccines, that much harder to persuade.

Faced with these accruing obstacles, doctors and others admit to bouts of “outreach fatigue,” exasperation and despair.

“I just don’t know what else I can do,” Dr. Wiltz said. “Some people you just can’t convince and you have to accept that’s the way it’s going to be.”

Some outreach campaigns are turning their focus to getting boosters to the homebound and nursing home patients, hoping that older vaccinated people, among the most in danger from the virus, are readily amenable to an additional shot.

“We’ve got to revisit the places that we prioritized first, which were our senior centers in neighborhoods with folks that maybe had access challenges and who weren’t going to get that initial vaccination easily,” said Dr. Jennifer Avegno, director of the New Orleans health department.

In Franklin, La., Dr. Wiltz, who serves a predominantly Black community, said that initial tireless discussions about the vaccine by doctors and ministers with older, reluctant patients were paying off in their enthusiasm for the boosters. “They don’t have to be convinced,” he said. “They’re already there.”

But while doctors try to encourage eligible vaccinated people to come in for boosters, they struggle to defend the need for the third shot to those who have yet to get their first. “Between boosters and the unvaccinated, it’s now really two different types of campaigns,” Dr. Avegno said.

Of late, many who are wary of the vaccine say they have become more confused by what they see as mixed messages from federal health agencies and the White House. To get them straightforward information, Yamhill County, Ore., will soon be offering discreet phone appointments. People who want to learn more about the shots will be able to sign up online and get a call from a local physician.

“People have lots of questions and they want a confidential way to have that conversation,” said Lindsey Manfrin, director of the county’s health and human services office. “Unfortunately, there’s stigma here around getting vaccinated and there’s stigma around not getting vaccinated.”

Health officials like Ms. Manfrin are reaching down deep to come up with creative solutions, and redoubling efforts to engage primary care providers and faith and business leaders to help them win over the holdouts, one by one.

Dan Mehan, president of the Missouri Chamber of Commerce and Industry, a pro-business group in a state where vaccination rates lag nationally, is providing companies a “vaccination encouragement” tip sheet and awarding them bronze, silver or gold-level certificates, based on percentage of vaccinated employees. Employers can then flaunt their status to encourage customer traffic. “We think vaccination is essential for the recovery from the pandemic,” Mr. Mehan said.

Grief-stricken relatives of children and unvaccinated adults who succumbed to the Delta variant have even been taking it upon themselves to sponsor vaccine drives. This summer, some held vaccine events at funerals.

But with mass vaccine sites largely shuttered, the burden of persuasion has fallen increasingly to primary care providers. Dr. David Priest, an infectious disease specialist with Novant Health, which has many clinics in North Carolina, has had repeated discussions with hesitant patients around the Covid vaccines.

“You have to overcommunicate to an incredible degree,” Dr. Priest said, “because we still get questions on things that I think, ‘This was well-known 18 months ago.’ But that’s where people are, so you just have to keep answering that question and answering it and answering it.”

It is critical, he added, that doctors have vaccines on hand. “So when the patient finally says, ‘I think I’ll do it,’ we can seal the deal. Because if you don’t have the shots in your clinic right then, people get in their car, get busy with other errands, forget or change their mind.”

Alison Buttenheim, a behavioral health expert at the University of Pennsylvania, noted that although primary care doctors, as trusted sources for patients, had been playing a crucial role in this phase of vaccine uptake, “it definitely raises the question of what happens to people who don’t have a usual source of care.”

But at this point, many doctors and nurses say they are exhausted by putting in so much persuasive effort, for so many months, with relatively little return, even as they are treating very ill patients who had refused to get vaccinated.

“It is an uphill battle,” said Dr. Uzma Syed, an infectious disease specialist in Jericho, N.Y., on Long Island, who for months has been giving vaccine education talks to national and international groups. “I can’t say that these conversations don’t come with tremendous burnout. But you keep going in hopes that you reach even one person to change their mind, because that’s a life saved.”