Demonstrators rally against anti-abortion and voter suppression laws at the Texas State Capitol on October 2, 2021 in Austin, Texas.
CNN  — 

An organization that operates several clinics in Texas said Thursday that its staff has provided abortions for patients who are more than six weeks into their pregnancies, after a federal judge Wednesday blocked a Texas law banning the procedure roughly after that point.

The organization, Whole Woman’s Health, announced in tweets and in a call with reporters that it had resumed the procedure after six weeks.

“Last night, we reached out to some of the patients that we had on the waiting list, to come in to have abortions today – folks whose pregnancies did have cardiac activity earlier in September,” Amy Hagstrom Miller, the president and founder of the organization, said on the call. “We were able to see a few people at 8, 9 this morning right away when we opened the clinic.

Under the Texas law, the clinics and their staff could potentially still be held liable in private state court litigation for those abortions if the judge’s order blocking the law is ultimately reversed by a higher court.

US District Judge Robert Pitman issued the order blocking the law Wednesday night in a lawsuit brought by the Justice Department last month.

Texas officials, as well as individuals who have intervened to defend the law, have already indicated they are appealing the decision to the 5th US Circuit Court of Appeals, but they have not yet formally asked the appeals court to put the order on hold.

Pitman said Wednesday that Texas had “contrived an unprecedented and transparent statutory scheme to” deprive its residents of their abortion rights.

Rather than task government officials with enforcing the ban, the Texas Legislature deputized private citizens to bring lawsuits in state court against abortion providers or anyone else who assists a person in obtaining an abortion after fetal cardiac activity is detected, a point usually around six weeks into a pregnancy and often before a woman knows she is pregnant.

The enforcement mechanism, by design, has complicated the usual route abortion rights advocates take in seeking court orders to block extreme restrictions of the procedure. Pitman issued a sweeping order that included prohibitions on state court officials, such as judges, from moving forward with state court litigation brought by private citizens under the law.

Previous attempts by clinics and others to obtain court orders, including at the 5th Circuit and the US Supreme Court, to block the law had been unsuccessful.

“From the moment S.B. 8 went into effect, women have been unlawfully prevented from exercising control over their lives in ways that are protected by the Constitution,” Pitman said in his order. “That other courts may find a way to avoid this conclusion is theirs to decide; this Court will not sanction one more day of this offensive deprivation of such an important right.”

Miller, the Whole Woman’s Health president, said Thursday that “some” of the organization’s physicians felt comfortable resuming the procedure after Pitman’s order came down and that a “large group of physicians” at the organization “are comfortable if this injunction is upheld by the Supreme Court.”

She said that the volume of phone calls to the organization increased Thursday. “There is actually hope from patients and from staff, and there is a little desperation in that hope. Folks know that this opportunity could be short-lived,” she said.

Texas Right to Life, an anti-abortion group that had advocated for the ban, has emphasized that if Pitman’s order is reversed, state court enforcement actions could be brought under the law against those who facilitated abortions after six week while the order was in place.

“The abortion industry is taking a significant risk of future liability by deciding to continue their violent business as usual,” Kimberlyn Schwartz, the group’s director of media and communication, said in an email to CNN. “Texas Right to Life intends to ensure the law is fully enforced when this case is settled. There is a four year statute of limitations, so the industry can be sued eventually for abortions performed today.”

This story has been updated with additional details Thursday.

CNN’s Ariane de Vogue contributed to this report.