An idea I had would be a NT-owned shuttle coming to the Cyberiad, and needing to be repaired before it could be sent off. It would dock where the Emergency shuttle would normally, be, and need to be rebuilt to repair any hull breaches, broken tech, and resupplied.
It could keep all of the departments busy, with Engineering repairing it, and Security keeping it secure. Science could need a certain R&D level to repair a computer on it, with mining bringing them the supplies to make it. And then Supply and Medical could be needed to stock it up before sending it out. Maybe even have an injured pilot need to be cared for?
Main issues I can think of, is that it would require an above-average amount of admin involvement, and might impede calling the Emergency Shuttle until it is sent off.