Federal prosecutors investigating former President Donald J. Trump’s handling of classified documents have obtained the confidential cooperation of a person who has worked for him at Mar-a-Lago, part of an intensifying effort to determine whether Mr. Trump ordered boxes containing sensitive material moved out of a storage room there as the government sought to recover it last year, multiple people familiar with the inquiry said.

Through a wave of new subpoenas and grand jury testimony, the Justice Department is moving aggressively to develop a fuller picture of how the documents Mr. Trump took with him from the White House were stored, who had access to them, how the security camera system at Mar-a-Lago works and what Mr. Trump told aides and his lawyers about what material he had and where it was, the people said.

At the heart of the inquiry is whether Mr. Trump sought to hide some documents after the Justice Department issued a subpoena last May demanding their return.

The existence of an insider witness, whose identity has not been disclosed, could be a significant step in the investigation, which is being overseen by Jack Smith, the special counsel appointed by Attorney General Merrick B. Garland. The witness is said to have provided investigators with a picture of the storage room where the material had been held. Little else is known about what prosecutors might have learned from the witness or when the witness first began to provide information to the prosecutors.

But prosecutors appear to be trying to fill in some gaps in their knowledge about the movement of the boxes, created in part by their handling of another potentially key witness, Mr. Trump’s valet, Walt Nauta. Prosecutors believe Mr. Nauta has failed to provide them with a full and accurate account of his role in any movement of boxes containing the classified documents.

In the past few weeks, at least four more Mar-a-Lago employees have been subpoenaed, along with another person who had visibility into Mr. Trump’s thinking when he first returned material to the National Archives, according to people briefed on the matter. Two people said that nearly everyone who works at Mar-a-Lago has been subpoenaed, and that some who serve in fairly obscure jobs have been asked back by investigators.

Prosecutors have also issued several subpoenas to Mr. Trump’s company, the Trump Organization, seeking additional surveillance footage from Mar-a-Lago, his residence and private club in Florida, people with knowledge of the matter said. While the footage could shed light on the movement of the boxes, prosecutors have questioned a number of witnesses about gaps in the footage, one of the people said.

But hoping to understand why some of the footage from the storage camera appears to be missing or unavailable — and whether that was a technological issue or something else — the prosecutors subpoenaed the software company that handles all of the surveillance footage for the Trump Organization, including at Mar-a-Lago.

And they recently subpoenaed Matthew Calamari Sr., the longtime head of security at the Trump Organization who became its chief operating officer. His son, Matthew Calamari Jr., who is the company’s corporate director of security, was subpoenaed some time ago, according to a person familiar with the activity.

Both would have insight into the security camera operation, according to people familiar with the matter. Both Calamaris appeared before the grand jury gathering evidence in the case on Thursday. CNN first reported that prosecutors planned to question them.

One of the previously unreported subpoenas to the Trump Organization sought records pertaining to Mr. Trump’s dealings with a Saudi-backed professional golf venture known as LIV Golf, which is holding tournaments at some of Mr. Trump’s golf resorts.

It is unclear what bearing Mr. Trump’s relationship with LIV Golf has on the broader investigation, but it suggests that the prosecutors are examining certain elements of Mr. Trump’s family business.

A spokesperson for Mr. Trump called the case “a targeted, politically motivated witch hunt” that is “concocted to meddle in an election and prevent the American people from returning him to the White House.” The spokesperson accused Mr. Smith’s office of harassing “anyone who has worked for President Trump” and of now using the inquiry to target Mr. Trump’s business.

Investigators have been piecing together Mr. Trump’s handling of government documents for months, seeking information not just about his habits after leaving the White House but also about his practices as president. Among the information they have gathered in interviews concerned his habit of flushing material down toilets, according to a person familiar with the matter.

Another related line of inquiry for Mr. Smith’s team is whether Mr. Trump misled one of his lawyers, M. Evan Corcoran, about the movement of classified documents around Mar-a-Lago. In June, Mr. Corcoran helped draft a sworn statement, signed by another lawyer, saying that a “diligent search” was conducted of the boxes and that any classified documents were turned over to the Justice Department.

To obtain Mr. Corcoran’s testimony on this subject, which would ordinarily be blocked by attorney-client privilege, the special counsel’s office first had to convince a judge that Mr. Trump may have misled him. In doing so, the prosecutors invoked what is known as the crime-fraud exception, which allows them to pierce attorney-client privilege when they have reason to believe that a client used legal advice or legal services in furthering a crime.

During his appearance before the grand jury in March, Mr. Corcoran testified that several Trump employees had told him that the Mar-a-Lago storage room was the only place where the documents were kept, according to people with knowledge of the matter. The employees turned out to be wrong — when F.B.I. agents searched Mar-a-Lago in August they found classified documents in Mr. Trump’s office and residence — but at the time, that was a common belief within Mr. Trump’s inner circle.

Although Mr. Corcoran testified that Mr. Trump did not personally convey that false information, his testimony hardly absolved the former president, the people with knowledge of the matter said. Mr. Corcoran also recounted to the grand jury how Mr. Trump did not tell his lawyers of any other locations where the documents were stored, which may have effectively misled the legal team.

Prosecutors working under Mr. Smith have developed what multiple people familiar with the investigation say is a wealth of testimony and evidence about Mr. Trump’s behavior during the lengthy period when the National Archives and the Justice Department sought to retrieve presidential materials from the former president.

After months of requests, Mr. Trump in January 2022 turned over to the archives 15 boxes of material he had taken from the White House. Those boxes turned out to contain reams of classified material, prompting a Justice Department investigation and a subpoena in May of last year demanding the return of any further documents in Mr. Trump’s possession.

Mr. Corcoran turned over another set of documents in response to the subpoena. But suspecting that Mr. Trump still had more based on witness testimony and video footage, prosecutors sought a search warrant, which the F.B.I. used to scour Mar-a-Lago in August, turning up more material despite the earlier statement from the lawyers saying they had found nothing else there.

The Justice Department investigation has returned repeatedly in recent weeks to a crucial question: Did Mr. Trump instruct Mr. Nauta, or anyone else, to move boxes out of the storage room before the lawyers conducted the “diligent search” of Mar-a-Lago and said no classified records remained at the property?

Last fall, prosecutors faced a critical decision after investigators felt Mr. Nauta had misled them. To gain Mr. Nauta’s cooperation, prosecutors could have used a carrot and negotiated with his lawyers, explaining that Mr. Nauta would face no legal consequences as long as he gave a thorough version of what had gone on behind closed doors at the property.

Or the prosecutors could have used a stick and wielded the specter of criminal charges to push — or even frighten — Mr. Nauta into telling them what they wanted to know.

The prosecutors went with the stick, telling Mr. Nauta’s lawyers that he was under investigation and they were considering charging him with a crime.

The move backfired, as Mr. Nauta’s lawyers more or less cut off communication with the government. The decision to take an aggressive posture toward Mr. Nauta prompted internal concerns within the Justice Department. Some investigators believed that top prosecutors, including Jay Bratt, the head of the counterespionage section of the national security division at the Justice Department, had mishandled Mr. Nauta and cut off a chance to win his voluntary cooperation.

More than six months later, prosecutors have still not charged Mr. Nauta or reached out to him to renew their conversation. Having gotten little from him as a witness, they are still seeking information from other witnesses about the movement of the boxes.

In interviews recently, the Justice Department has been focused on Mr. Nauta and the help he received from a Mar-a-Lago maintenance worker in moving boxes. They have asked multiple people questions about it, as well as questions about the security cameras and what they did and did not capture. They have asked questions specifically about whether Mr. Nauta was walking to or from the president’s residence on the property, according to a person briefed on the matter.

In addition to seeking testimony from the Calamaris and other Trump Organization employees, the special counsel’s office has issued numerous subpoenas to the company itself, seeking a variety of internal documents, according to people with knowledge of the subpoenas.

Another line of inquiry that prosecutors have been pursuing relates to how Mr. Trump’s aides have helped hire and pay for lawyers representing some of the witnesses in investigations related to the former president. They have been trying to assess whether the witnesses were sized up for how much loyalty they might have to Mr. Trump as a condition of providing assistance, according to people briefed on the matter.

William K. Rashbaum contributed reporting.