Why did Iran's government waited until jimmy carter left office to release the hostages?

This had little to do with Jimmy Carter and more to do with the policies of the state.

Explanation:

The state of Iran had recently turned into a Islamic one after the revolution and their basis as a nation had been the rule of their theological leader.

They had been wary of the US and did not want to risk a war but also did not want to be smothered by them.

So, the policy was to keep the hostages until a regime change for a proper deadlock between the two nations and avoid direct conflict

Once that was done the hostages were released.