William F. Nolan wrote two sequels to Logan’s Run: Logan’s World and Logan’s Search. There is also a novelette, Logan’s Return, that has been published as an e-book.
Logan’s World deals with events following Logan’s returning to Earth, amidst the survivors and ruins of the system he escaped in the first novel, while Logan’s Search deals with Logan going to an alternate reality (with the assistance of aliens) to once again stop the government system he escaped in the first novel, albeit with some minor changes.