AUSTIN, Texas—Women’s health clinics in Louisiana, Oklahoma and beyond are reporting an influx of out-of-state patients from Texas looking for abortions and other services, weeks after the U.S. Supreme Court allowed Texas’s restrictive abortion law to take effect.

Some clinics in neighboring states said they are overwhelmed with out-of-state patients. They are adding hours to try to handle the backlog, administrators said. In Texas, meanwhile, clinics are turning most of their patients away and losing their staff.

The...

AUSTIN, Texas—Women’s health clinics in Louisiana, Oklahoma and beyond are reporting an influx of out-of-state patients from Texas looking for abortions and other services, weeks after the U.S. Supreme Court allowed Texas’s restrictive abortion law to take effect.

Some clinics in neighboring states said they are overwhelmed with out-of-state patients. They are adding hours to try to handle the backlog, administrators said. In Texas, meanwhile, clinics are turning most of their patients away and losing their staff.

The migration out of state follows the enactment of the Texas Heartbeat Act, which became effective on Sept. 1 after the Supreme Court turned away a legal challenge.

The law bans abortions after “cardiac activity” can be detected, usually around six weeks of gestation, and it deputizes private citizens to sue anyone they believe may have aided such a procedure and collect $10,000. On Monday, a Texas doctor who publicly said he performed an abortion was sued in state court by two different plaintiffs, handing Texas the first tests of the law.

In Shreveport, La., about 200 miles east of Dallas, staff members at Kathaleen Pittman are booked more than three weeks out and have begun adding some evening hours to get patients in for appointments as quickly as possible, said administrator Kathaleen Pittman.

“Because of the delay, they’re going to be farther along,” Ms. Pittman said of the effect of the new Texas law. Most patients she sees report their reason for terminating a pregnancy is financial, she said, and many are struggling with the cost of travel to Louisiana and taking time off work for the two required appointments.

In Tulsa, Okla., at least 70% of patients at Tulsa Women’s Clinic are now from Texas, said executive administrator Andrea Gallegos. Some are coming from as far as the Rio Grande Valley, a 12-hour drive away. The clinic has tripled its number of daily appointments to about 35 and added some Saturday hours, Ms. Gallegos said.

The average Texas woman of childbearing age is 17 miles from an abortion clinic in-state, but 247 miles from one out-of-state, according to an analysis from the Guttmacher Institute, a research group supportive of abortion rights.

Joe Pojman, executive director of Texas Alliance for Life, a group that advocated for the Texas law, said the state has increased funding by $20 million to assist women who carry pregnancies to term. “Texas has vast resources to help a woman so that no woman seeks an abortion in Texas or out of state because she has no alternatives,” he said.

Mr. Pojman said at least 20 of the state’s 23 licensed abortion providers are operating, but his organization doesn’t know how many abortions are being performed.

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Clinics in Texas report that many staff members have quit since the law took effect, saying they are worried about the possibility of being sued, even if they complied fully with the new law. “My staff are scared to come to work. I usually have eight. I have two or three today,” said Linda Shafer, administrator of Aaron Women’s Clinic in Houston.

Amy Hagstrom Miller, founder of Whole Woman’s Health, which operates four clinics in North, Central and South Texas, said the clinics are now denying abortions to an estimated 80% of patients they have seen since the new law went into effect, turning them away after an ultrasound. The most the clinics can do at that point is share with patients a few websites listing other options, Ms. Miller said. “They’re leaving the clinic without a plan,” she said of patients.

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Abortion providers and care advocates said that more women are turning to self-administered methods, like using abortion pills ordered online. Ms. Miller said most of Texas’s abortion clinics are small, independent clinics that may be forced to close permanently if the law remains in place.

One of the areas most affected is the Rio Grande Valley, the Texas region farthest from any other state and hemmed in by Mexico to the south and immigration checkpoints to the north. That prevents anyone in the country illegally from traveling for an abortion, said Lucy Felix, a Brownsville-based outreach coordinator for the National Latina Institute for Reproductive Health. “For our undocumented immigrant communities, forced pregnancy is now the law of the state,” she said.

In addition to the suits filed Monday, the U.S. Justice Department has sued Texas seeking to block the law, with a hearing set for Oct. 1.

Write to Elizabeth Findell at Elizabeth.Findell@wsj.com