UK MP representing jailed British-Iranian mother calls for meeting with new foreign secretary
Tulip Siddiq, MP for Hampstead and Kilburn, described Hunt's predecessor Boris Johnson's record as "one of failure".
He added that Johnson - who resigned as foreign minister on Monday - had "actively damaged" the case by wrongly claiming Zaghari-Ratcliffe was in Iran training journalists.
Zaghari-Ratcliffe, who works for the Thomson Reuters Foundation, is serving a five-year prison sentence for allegedly planning the "soft toppling" of Iran's government while traveling with her young daughter.
The young girl, Gabriella, has remained in Iran in the care of relatives.
Last month, her husband said she now faces a new charge of "spreading propaganda against the regime".
He added that "the judge told Nazanin to expect that there will likely be another conviction and sentence against her".
The case has become highly politicised, especially after a careless blunder by the UK Foreign Secretary Boris Johnson in December.
Johnson mistakenly claimed Zaghari-Ratcliffe had been training journalists in Iran - something the Thomson Reuters Foundation and her family have strongly denied.
This has been used by the Iranian authorities to help justify the new charges and may have contributed to her lengthened sentence.
Hunt was appointed foreign secretary on Monday after Johnson submitted his resignation in light of straggling Brexit talks.
In a letter shared by BuzzFeed, Siddiq told Hunt: "The very minimum one expects from the Foreign Office is that it seeks to protect British citizens who are in distress abroad.
"I'm afraid in that regard, your predecessor's record was one of failure.
"Despite his welcome promise to 'leave no stone unturned' for my constituent – he failed to deliver any meaningful progress to secure her release.
"His comments that she was in Iran to 'teach journalism' were baseless and actively damaged her case."
Siddiq added she was keen to understand "what issues are blocking progress and what steps you will take to accelerate efforts to bring Nazanin back to West Hampstead".
Zaghari-Ratcliffe's case comes as London is considering repaying Tehran some 400 million British pounds from a pre-1979 arms deal.