The new PM has so far rejected demands to launch an investigation after he shared sensitive documents with Tory backbenchers from private emails without permission.
Labor has urged the government to publish its assessment of Suella Braverman's security breaches following her controversial reappointment as interior minister just six days after she was forced out.
New Prime Minister Rishi Sunak is under increasing pressure to reinstate Braverman as interior secretary after a former party chief claimed he had committed "several violations" of the ministerial code.
He has so far rejected demands to launch an investigation after he shared sensitive documents with Tory backbenchers from private emails without permission.
Both Labor and the Liberal Democrats have raised "national security" concerns over the abuses, and called for a Cabinet Office investigation.
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Labor will try to encourage ministers to share this risk assessment and other alleged leaks, as well as information provided to Sunak before he reposted it at the Home Office, with a "humble address" motion in Parliament.
Braverman has so far refused to appear before lawmakers to explain the circumstances surrounding what happened.
Shadow Home Secretary Yvette Cooper said: "Rishi Sunak and Suella Braverman cannot keep running away from these questions. This is too serious for that, and raises serious doubts about the prime minister's judgment.
"People need to know that they can trust the Minister of Home Affairs with very sensitive information and our national security. Rishi Sunak's decision to reappoint Suella Braverman is highly irresponsible.
"Labour will use every open parliamentary mechanism to force the government to come clean on his re-appointment, to get answers and request detailed documents to be released to the Intelligence & Security Committee."
Braverman resigned from his post more than a week ago after using his personal email address to forward sensitive government documents, violating rules that ministers must abide by.
But the new prime minister put him back in the Home Office on Tuesday, and backed the decision after being pressed in the Commons on Wednesday.
Labor leader Sir Keir Starmer has called for the interior minister to be sacked and accused Sunak of brokering a "dirty trade deal for support" in the Tory leadership contest, which he won after receiving Ms Braverman's support.
But Sunak insisted the interior minister had "learned from his mistakes" and that he did not regret his decision to reappoint him - although some Conservative lawmakers added their voices to the backlash.
Conservative MP Caroline Nokes said there were "big questions hanging over this whole issue" and supported opposition calls for an investigation, while former Tory chair Sir Jake Berry described the abuses as "very serious".
Shadow secretary of state David Lammy called for a full investigation into Braverman's actions and for him to be fired, telling Sky News: "Secretary of the interior is the most serious job you can have in our state.
"This is a person who makes judgments about terrorism and counter-terrorism, who makes judgments about very, very serious offenders, whether they should be allowed out of prison, and for that reason, someone who, I'm afraid, judgment is very important.
"I fear it was a misjudgment that, indeed, he was fired. The question is, why was he brought back?"
Nicknamed "Leaky Sue", Braverman was reportedly previously investigated by government officials after the leak of a story involving the security services.