Texas police departments expressed confusion over how to proceed.

Brad Coe, the sheriff in Kinney County along the Texas border, woke up on Wednesday with one destination in mind: Gov. Greg Abbott’s office.

Sheriff Coe is the lead police authority in a mostly rural county that has embraced the state’s attempt to halt the sharp rise in migrants coming from Mexico. He said he was done with the whiplash of confusing legal orders issued in the past 24 hours over the law.

He wanted to hear from Gov. Abbott himself. “I’m on my way to his office right now,” he said.

He recapped the conflicting directives handed down on whether his department had the legal authority to arrest migrants who have entered the country illegally, under the terms of the new law.

The Supreme Court ruled “that yes we can,” he said, referring to Tuesday’s decision. “But a lower court said no we can’t. The Supreme Court is supposed to be supreme court of the land. Something ain’t right.”

Police departments across Texas, both close to and far from the border, expressed confusion over how to proceed after the Supreme Court briefly allowed the new law to go into effect on Tuesday — and then an appeals court once again halted it.

On Wednesday morning, a panel of three judges from the U.S. Court of Appeals for the Fifth Circuit heard more arguments from both the state and federal government about whether it should go into effect. Many police departments said they would be paying close attention to how the panel ruled before they unveil their plans.

The law, known as Senate Bill 4, would make it a state crime to cross into Texas from Mexico anywhere other than a legal port of entry. The first arrest would be considered a misdemeanor and after that, it would be a felony.

The state police have not given any public indication as to how or when it would begin enforcing the law if it ultimately goes into effect. Abril Luna, a spokeswoman with the police department in Brownsville, on the border, said that whatever the ruling, she did not expect daily operations to change dramatically. “If the law passes, of course it will be enforced just like any other Texas laws,” Ms. Luna said.

Jodi Silva, a spokeswoman for the police in Houston, the largest city in Texas, said they too would decide what course of action to take after a new court ruling is issued. “We are also monitoring and see which direction we go,” Ms. Silva said.

In San Antonio, Javier Salazar, the sheriff of Bexar County, which includes that city, issued a policy manual that directs his deputies to enforce the law without engaging in racial profiling, which many critics of the law fear would happen as state and local police try to determine who may have entered the country illegally. Deputies making arrests will base them “on probable cause supporting the elements of the offense and not on national origin, immigration status, ethnicity or race,” it said.

Back in Kinney County, Chief Coe, whose office employs six deputies, said that the long list of aggressive border security measures adopted by Mr. Abbott in recent months, which include placing concertina wire along the Rio Grande and a heavily armed law enforcement presence on parts of the border, appears to have deterred people from crossing. Last year, his deputies arrested up to 20 migrants a day. More recently, he said, “we’re lucky if we get two or three a day max.”

He said he hoped S.B. 4 would be allowed to take effect, not because he wants to arrest migrants, but because it would act as a deterrent. “People would stop coming because it is against the law,” he said.