High-powered immigrant advocacy organizations filed a lawsuit Wednesday to try to kick-start President Obama’s deportation amnesty, arguing that the Texas judge who halted the program two years ago can’t bind the rest of the country to his decision.

The groups filed a challenge in Illinois, building on a similar lawsuit filed over the summer in New York. If either one is successful, it could ignite a stampede of copycats, creating a patchwork across the country where Homeland Security would have to issue work permits and stays of deportation to some illegal immigrants, but would be barred from approving applications from someone just over the line in another court’s jurisdiction.

The Obama administration declined to comment, citing the pending case. But it is already fighting back in the New York lawsuit, arguing it has to follow the nationwide injunction imposed by Judge Andrew S. Hanen in February 2015 and upheld twice by a federal appeals court.

That ruling was sustained when the Supreme Court deadlocked, 4-4, in June.

Immigration advocates, however, say that since the tie ruling doesn’t create a national precedent, illegal immigrants can ask lower courts to take a look at the issue and decide whether they’ll be bound by Judge Hanen and the Fifth Circuit Court of Appeals.

“Their argument is potent but simple: What gives a federal judge in Brownsville, Texas, the authority to make decisions that alter their lives?” said David Leopold, a past president of the American Immigration Lawyers Association.

Michael Hethmon, a lawyer at the Immigration Reform Law Institute, said the challenge is not so much about immigration as it is about the legal boundaries of the judge’s ruling.

He said Judge Hanen’s ruling applied to Homeland Security, binding all of its officers, and arguing that whether or not the department should treat some applicants differently based on their location is tricky.

“They’re going to have an uphill battle,” he said.

Mr. Obama’s 2014 amnesty policy would grant work permits and a three-year stay of deportation to as many as 5 million illegal immigrants. He said it was legal under his powers of prosecutorial discretion: He wasn’t going to deport them, so he said it made sense to give them a tentative legal status.

The plaintiff in Wednesday’s lawsuit is Jose Lopez, a Mexican who was brought to Chicago when he was 4, and whose parents are illegal immigrants but whose two younger brothers are U.S. citizens.

Mr. Lopez applied for a two-year amnesty under a 2012 amnesty for so-called Dreamers, or young adult illegal immigrants. But his application was still pending when Mr. Obama announced his 2014 amnesty, which expanded the Dreamer program to three years.

Judge Hanen issued his injunction on Feb. 16, 2015, but the government — defying his order in what officials say was a technological snafu — approved and issued a three-year permit to Mr. Lopez and thousands of other applicants anyway.

One of those was Martin Jonathan Batalla Vidal, the plaintiff in the New York case, who filed his challenge over the summer.

Homeland Security has since revoked the permits of Mr. Vidal, Mr. Lopez and the others it erroneously approved, and reissued two-year permits to them instead.

Mr. Lopez argues that since he doesn’t live in Texas, and Illinois didn’t join the challenge to the amnesty, he shouldn’t be affected by Judge Hanen.

Mr. Batalla Vidal said in his court challenge that changing his three-year amnesty to a two-year amnesty means he’ll have to shell out $465 in fees to renew a year earlier. He also said the lack of a third year adds to uncertainty around his school and work plans.

One hurdle for the illegal immigrants is the terms laid out by Homeland Security for the amnesty. The government said it had sole discretion and nobody had a legal right to approval.

At the time, that was intended to be a defense against arguments that Mr. Obama had created a new blanket policy as opposed to exercising existing discretionary powers. But now it could trip the illegal immigrants.

The Obama administration had asked the Supreme Court to reconsider its 4-4 ruling, but the justices last week refused.

That sparked howls of protest from Latino rights groups, who said they’ll take the issue to voters in November, asking them to punish Republicans for failing to tackle the immigration issue head-on in Congress.

GOP presidential nominee Donald Trump has said he would revoke the 2012 amnesty, pulling tentative legal status from nearly 750,000 young adult illegal immigrants who have qualified. Mr. Trump has also vowed to stiffen enforcement of existing laws and to triple the size of the deportation agency, which could result in sending far more illegal immigrants back home.

Democratic opponent Hillary Clinton, meanwhile, has said she’ll try to go even further than Mr. Obama, flexing executive powers to expand his proposed deportation amnesty to nearly all illegal immigrants.

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