Questions about his travel have followed Mr. Long for days, as he sought to project confidence in the Trump administration’s response to the slow-moving storm. On Sunday, Mr. Long pushed back on reports that Kirstjen Nielsen, the secretary of homeland security, had asked him to step down because of his travel.

“Let me go ahead and clear up all the news: Secretary Nielsen has never asked me to resign,” Mr. Long said on NBC’s “Meet the Press.” “We have a very functional and professional relationship. We talk every day. We are both totally focused on Florence.”

He argued that the use of agency vehicles was appropriate given his “very critical and important role to make sure this government works on the nation’s worst day through continuity in government.”

“These vehicles are designed to provide secure communications and the program was actually developed in 2008 — it ran for me the same way it’s run for anybody else,” he said. “And you know, it’s my understanding that maybe some policies were not developed around these vehicles.”

Mr. Gowdy gave Mr. Long until Oct. 1 to produce relevant documents, and an aide said that Mr. Gowdy would wait until that date to proceed with other potential investigative actions.

Mr. Gowdy, who is retiring at the end of the year, has used his gavel selectively since taking over the Oversight Committee last year. But he considers excessive spending, particularly on executive branch travel, to be an issue of concern. He previously studied official travel taken by Scott Pruitt, the former administrator of the Environmental Protection Agency. And amid a rash of reports last fall about Trump administration officials’ use of private planes, Mr. Gowdy and Representative Elijah Cummings of Maryland, the committee’s top Democrat, jointly investigated travel by senior officials across the federal government.

Mr. Cummings said Monday that he supported an investigation but called on Republicans to simultaneously open an inquiry into the federal response to last year’s deadly hurricanes in Puerto Rico and the United States Virgin Islands. Committee Democrats issued a report this month accusing the Republicans of blocking a “credible” investigation.

“If Chairman Gowdy wants to do an investigation of FEMA, the priority should be on helping the people of Puerto Rico after the devastating hurricanes last year and ensuring that they are not treated like second-class citizens,” he said. “The committee should investigate the administrator’s use of government vehicles, but the more urgent focus right now should be on obtaining documents from the White House regarding its response to the hurricanes that — so far — the committee has refused to even request.”