WASHINGTON—The Justice Department moved quickly on Friday to appeal a federal judge’s latest ruling trimming President Donald Trump’s temporary travel ban, while the administration is separately laying the groundwork for new and broader restrictions.

The latest chapter in the fast-moving saga began Thursday evening in Honolulu, where U.S. District Judge Derrick Watson ruled that the Trump administration’s recent implementation of its ban contradicted the interim rules put in place by the Supreme Court on June 26.

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