A federal trade court has ordered Rodney S. Scott, the head of U.S. Customs and Border Protection, to appear at a hearing next month on the Trump administration’s handling of roughly $166 billion in tariff refunds.

The unexpected demand, issued on Wednesday, hinted at a judge’s ongoing concern that the government has not fully complied with a directive to return all of the money amassed under duties that were declared illegal by the Supreme Court earlier this year.

At the heart of the matter are the so-called reciprocal tariffs that President Trump previously applied to imports from around the world. In February, a majority of justices on the nation’s highest court struck down those tariffs, finding that Mr. Trump had exceeded his authority under law. That decision forced the government to repay the money, plus interest.

Within days of the Supreme Court’s ruling, the Court of International Trade took the first, formal steps to compel the administration to begin the refund process. While Mr. Trump publicly opposed returning the money, his administration still worked to set up a system to do so. It officially began accepting importers’ requests for repayment in late April.

But the refund system does not allow all importers to recover the taxes they paid from tariffs imposed using the International Emergency Economic Powers Act, or IEEPA. The government said in a court filing in April that it could process refunds for only about $127 billion out of the approximately $166 billion collected.

Otherwise, customs officials have said little publicly about how they might handle the remaining imports that were subject to IEEPA tariffs. The affected goods include those on which tariff payments were calculated and finalized earlier in Mr. Trump’s trade war, and others that are subject to more complicated import rules, the government has previously indicated.

The lack of clarity prompted Richard K. Eaton, a judge on the trade court, on Wednesday to order a hearing for June, at which he directed Mr. Scott to “appear to answer the court’s questions as to the anticipated timing of Customs’ compliance with the court’s order.”

In a separate filing, the judge also cited concern about “millions” of entries for which the government has “not presented a proposal” for providing refunds.

A spokeswoman for Customs and Border Protection said the agency was reviewing the court order in consultation with the Department of Justice.

Ted Murphy, a top trade lawyer at the law firm Sidley Austin, said it was “extraordinary” for the court to order the head of an agency to testify in such a manner. But he said it was highly suggestive that the court had grown frustrated with the Trump administration, after months of patience over the issue of tariff refunds.

Mr. Murphy said that the trade court had been clear from the start that the government must “give back everyone their money.” But the administration, he said, has set up a system in which many importers had to apply, and still others potentially had to sue, just to reclaim what they paid.

“The court and the government are far apart,” Mr. Murphy said. “That’s what this boils down to.”

U.S. Customs and Border Protection did not immediately respond to a request for comment.

The court’s action could presage another round of intense legal wrangling between the U.S. government and importers over the extent of the president’s trade powers. So far, judges on the highly specialized trade court have repeatedly found against Mr. Trump, most recently ruling that his replacement tariffs — a 10 percent rate imposed on most foreign goods — violated the law. That case is on appeal and, if Mr. Trump loses, could add to the amount owed in refunds.

“We look forward to going back to court and making clear once again that these IEEPA tariffs were illegal and should be refunded immediately,” said Sara Albrecht, the chairman and chief executive of the Liberty Justice Center, which represented small businesses that had sued over the president’s tariffs.

Ana Swanson contributed reporting.