The morning after the leak of a draft Supreme Court opinion in a case concerning Mississippi’s only abortion provider, ​​Derenda Hancock arrived at the clinic around sunrise, just as she does most days, to lead a group of volunteers escorting patients. Wednesday morning, the group was back to do it again.

Women regularly come to the clinic, Jackson Women’s Health, from across Mississippi, as well as Louisiana and Texas. This week, at least one came from Tennessee.

The clinic challenged the Mississippi law that would shut down abortions there, in a case that has climbed to the Supreme Court and paved the way to what will surely be one of the most consequential decisions on abortion rights in decades.

Often, it can seem like a wide gulf separates the legal fight in Washington from the practical realities on the ground. But the draft opinion has brought renewed concern about the fate of the clinic once the final ruling arrives.

“We know there is no winning,” Ms. Hancock said by phone as she helped direct traffic and kept watch over the perimeter of the clinic, known as the Pink House for its flamingo exterior. The volunteers that she leads go by the name Pink House Defenders.

For now, the mind-set at the clinic, among doctors, staff members and volunteers, was to keep pushing ahead for as long as they could.

“We’re going to continue to see patients,” said Shannon Brewer, the director of the clinic. “I don’t know anything different we would do other than see patients. It’s a surprise to some people for some reason. This is what we’ve anticipated.”

The clinic has seen a surge in demand in recent months, driven in large part by the rules implemented in Texas that are some of the most restrictive in the nation, pushing people seeking an abortion to surrounding states. The clinic went from operating three days a week to five. Some months, the clinic has as many as 300 patients.

“I don’t recall ever having to turn someone away because we didn’t have a slot for them until the past couple of months,” said Dr. Cheryl Hamlin, an obstetrician-gynecologist from Massachusetts who travels to Jackson once a month, part of an out-of-state rotation of doctors who keep the clinic operating. It has been unable to find physicians in Mississippi willing to provide abortions.

There had been some discussion for plans if Roe v. Wade was overturned, such as finding ways to help women in Mississippi travel to other states to obtain an abortion. Those patients, Ms. Brewer said, would be the ones most severely impacted by the ruling. Ms. Brewer has also talked about moving the clinic to New Mexico, if necessary.

But mostly, she wants to keep the Jackson clinic open for as long as she can. The clinic has not had a staff doctor for more than a decade. Over the years, other providers in Mississippi shuttered, buckling under the restrictions imposed by the state. The Jackson clinic abides by them, though, providing ultrasounds and mandated counseling.

Dr. Hamlin also tells patients that abortion creates a heightened risk of breast cancer, just like the law dictates, even though it is a claim that experts say is unsupported by science.

At the end of her counseling sessions, in a quick dash of defiance, Dr. Hamlin urges her patients to vote, if they believe that access to abortion is important.

“I’m going to be more forceful,” she said of her plea. “What have you got to lose?”