Actually according to a documentary, the Iranians held on to the hostages and the announcement of their impending release just long enough to bring down the Carter presidency. It wasn’t until Reagan was officially sworn in and then basically the Iranians announced that the hostages would be brought back to the US.
The Carter Administration asked to continue trying to negotiate the release of the hostages after Carter had lost the elections. The Iranians drag it out, trying to get the best deal they could from the weaklings and incompetents in the Carter administration.
The Iranians did not want to go up against Reagan.
At the end the Iranians accepted Carters offer of money and other concessions to release the hostages.
Many thought Reagan should simply refuse to honor the deal after the hostages were released, saying it was extortion and ransom and therefore an illegal deal.