The desired features beyond the minimum set are accepted as being at risk of remaining unimplemented if the due date can't be shifted further to the right. They become stretch goals that may be achieved by the deadline, or will be worked into a second effort during the maintenance of the system.
Agile (especially some of the stuff in the Lean Software world) is very well-suited to this kind of development. As are the classical evolutionary or iterative & incremental models.