It was a shocking double victory for the rioters who attacked the Capitol on Jan. 6, 2021, when President Trump granted them clemency for their crimes on his first day back in the White House and then, in the months that followed, allowed his Justice Department to purge many of the federal agents and prosecutors who sought to hold them accountable.

But even though the president has given them their freedom and has taken steps toward satisfying their desire for retribution, they are asking for more: In the past several weeks, the rioters and their lawyers have made a concerted effort to push the Trump administration into paying them restitution for what they believe to be their unfair prosecutions.

On Thursday, one of the lawyers, Mark McCloskey, said during a public meeting on social media that he had recently met with top officials at the Justice Department and pitched them on a plan to create a special panel that would dole out financial damages to the rioters — much like the arrangement of a special master to award money to the victims of the Sept. 11, 2001, terrorist attacks.

The panel, which Mr. McCloskey called a “voluntary nonjudicial resolution committee,” would consider rioters’ cases individually, he said, then assign them sums according to harms they had purportedly suffered at the hands of the federal government.

Mr. McCloskey said that he wanted the panel to be overseen by Jeanine Pirro, who runs the federal prosecutors’ office in Washington that took the lead in filing charges against nearly 1,600 rioters who joined in the Capitol attack.

“The only thing I can do as your lawyer,” he told the rioters who were at the online meeting, “is to turn your losses into dollar bills.”

Neither Ms. Pirro nor a spokesman for the Justice Department responded on Sunday to messages seeking comment on Mr. McCloskey’s plan, and it remains unclear how seriously top administration officials are taking it.

Still, the proposal represented a serious escalation of the efforts to rewrite the history of Jan. 6. If accepted, it would effectively designate members of the mob, whose Capitol break-in upset the lawful transfer of presidential power, as victims of the government and deserving of reparations.

Mr. McCloskey, who rose to prominence five years ago after he pointed an AR-15-style rifle at social justice protesters outside his home in St. Louis, has been leading the efforts to secure restitution for the rioters since at least March, when he announced that he and another lawyer, Peter Ticktin, a former classmate and longtime ally of Mr. Trump’s, were planning to sue the government.

The men originally said they were intending to bring claims that the people who were prosecuted for taking part in the Capitol attack had been mistreated by federal agencies like the Justice Department and the Bureau of Prisons.

In May, many of the rioters took heart when Mr. Trump’s Justice Department agreed to pay nearly $5 million to settle a separate legal action: a wrongful-death lawsuit that was brought during the Biden administration by the family of Ashli Babbitt, an Air Force veteran who was fatally shot by the police while she was in a crowd that was trying to break onto the House floor on Jan. 6.

The settlement raised hopes that the department might look with favor on other lawsuits filed by the rioters themselves. But late last month, those hopes were tempered after department lawyers formally opposed a lawsuit that was brought in June by five members of the Proud Boys who were charged with sedition in connection with Jan. 6. The lawsuit claimed that federal officials had subjected them to “political persecution” as “allies of President Trump.”

During the online meeting last week, Mr. McCloskey acknowledged that he and Mr. Ticktin had also run into “significant difficulties” in pursuing legal action on behalf of the rioters.

He acknowledged that there could be problems following through on his initial plan to file cases under the Federal Tort Claims Act, which allows individuals to sue the government for injuries caused by federal employees. He also said it could be challenging to overcome the two-year statute of limitations on bringing tort claims against the government for things that happened nearly five years ago.

But Mr. McCloskey assured the rioters that they had allies inside Mr. Trump’s Justice Department. Chief among them, he said, was Ed Martin, who runs the so-called weaponization working group, a body that was created to investigate those who investigated Jan. 6 and other people whom Mr. Trump perceives to be his enemies.

“He’s 100 percent on our side,” Mr. McCloskey said of Mr. Martin.

Mr. Trump’s grant of clemency to the Jan. 6 defendants was one of the most remarkable uses of presidential mercy in modern history. But also remarkable is the extent to which many of the rioters have remained unsatisfied by the measure, as well as by the subsequent firings and demotions of more than two dozen federal prosecutors and F.B.I. agents who worked on Capitol riot cases.

On Saturday, for example, Enrique Tarrio, the leader of the Proud Boys who was freed by Mr. Trump from a 22-year prison term stemming from Jan. 6, posted what amounted to a list of demands to the administration in a social media message. Among the things he called for were compensation for the rioters “for their suffering and that of their families” and the firing of “everyone involved” in the riot cases.

“If this isn’t done,” Mr. Tarrio wrote, “we will all hang together.”

On Sunday, another rioter, Ryan Nichols, a former Marine who was sentenced to more than five years in prison for joining a crowd that shoved at officers in a tunnel outside the Capitol, doubled down on his attacks against the police in a post on social media.

“I’d do it again given the same situation,” Mr. Nichols said. “They attacked Americans and killed innocent protesters.” He added that “we should have” dragged foes “through the streets.”

Four people in the mob died on Jan. 6, 2021, including Ms. Babbitt; two died by natural causes and another from an overdose. Five officers died in the days and weeks after the attack, including two suicides.

One rioter, Shane Jenkins, who was sentenced to 84 months in prison for assaulting an officer and shattering a window at the Capitol with a tomahawk on Jan. 6, spoke during the online meeting and captured the spirit of loss and disillusionment that many of the pardoned defendants seem to feel.

Mr. Jenkins compared the rioters to the biblical story of the Israelites who were enslaved and then released by God from bondage in Egypt, only to roam for decades through the desert.

“Through Trump, God pardoned us and set us free, right?” Mr. Jenkins said. “Well, then what did they do? They wandered around the desert for 40 years and I don’t think very many of them got to see the Promised Land.”

“I just feel,” he went on, “like that’s kind of where we’re at right now.”