You have to direct your EMT to collect a stretcher from the ambulance, then to use that stretcher to pick up the patient and put him in the ambulance. If you do that, when he gets to the hospital he'll unload the pt into the hospital, then come back out and get in the truck.
Alternatively, just always send your ambulance back to HQ instead of hospital, then it doesn't matter how they're loaded and you'll still get the points for completing the task.