On Thursday evening, the Supreme Court temporarily blocked a law that would have restricted abortion rights in Louisiana. The case was the first abortion-related decision that Justice Brett Kavanaugh has decided on since joining the court last year. And abortion rights groups closely followed the ruling.

Even though Kavanaugh wrote the dissenting opinion, those groups didn't get a clear look into how the new judge may view abortion issues in the future, said Travis Tu, lead counsel on the case for the Center for Reproductive Rights.

"Justice Kavanaugh plays his cards very close to the vest in this ruling," Tu said in a telephone interview with CBS News. "This is just one step in a long game."

Tu noted that the only precedent that Kavanaugh referenced in his dissent was Whole Woman's Health v. Hellerstedt, a 2016 Supreme Court case that said states weren't allowed to place an "undue burden" women seeking an abortions.

In the second page of his decision, Justice Kavanaugh acknowledged the ruling: "All parties, including the State of Louisiana, agree that Whole Woman's Health is the governing precedent for purposes of this stay application," he wrote. "I therefore will analyze the stay application under that precedent."

But, according to Tu, "it's not clear from his writing whether he supports that decision or if he thinks it was wrongly decided."

In fact, although the decision was 5-4, Kavanaugh wrote the dissent separately, saying he would allow the law to take affect. He wrote he was prepared to reverse course if it did in fact burden a woman's access to abortion, as opponents claimed.

In Thursday's decision, justices were asked to look at a Louisiana law that requires doctors at abortion clinics to have admitting privileges at a nearby hospital. Admitting privileges allow doctors to directly admit their patients into a hospital, bypassing what can be a time-consuming check-in process.

Tu described the arrangement as more "commercial" than medical: Doctors promise to refer a set number of patients to a specific hospital in exchange for the ability to make that process simpler for their clients.

But abortion care doctors would rarely be referring business to a hospital because of the nature of the procedure, Tu said. Abortions are largely safe for women and it's a procedure that doesn't require intensive follow-up, like a knee or back surgery.

"While some doctors could promise 10 patients a year in exchange for admitting privileges, an abortion provider would struggle to refer 10 patients in their entire career," Tu said. Only one abortion care provider in the state has been successful in obtaining the privileges.

Because so few abortion providers receive these privileges, the policy would effectively close all but one of the state's few abortion clinics, Tu said. Some women, especially from the southern part of the state, would have to travel hours to obtain a procedure — an "undue burden," say pro-abortion rights groups.

In his dissent, Kavanaugh cited the law's 45-day window for doctors to obtain the admitting privileges to stay within the law.

"If the doctors, after good-faith efforts during the 45-day period, cannot obtain admitting privileges, then the [court's] factual predictions… could turn out to be accurately applied," Kavanaugh wrote in the third page of his decision.

"Justice Kavanaugh seems to think that doctors, if they tried hard enough, they might be able to get admitting privileges in 45 days, but that's just not how that works," Tu said. "These doctors have been trying to get admitting privileges for four and a half years, but hospitals have no incentive to give those privileges to doctors who provide abortion care."

The ruling is only a temporary measure, meaning it could set up a larger showdown if the court decides to hear this case in the upcoming term.