A carrier latch fails at the worst possible moment, and your cat is loose. What you should do next depends entirely on which side of the car door she is on. Inside the vehicle she is frightened but contained, and time is on your side. Outside, in a car park, she is one panic away from disappearing, and your behaviour in the next two minutes decides which way it goes.
Inside the Car: Keep It Sealed and Stay Calm
With a loose cat inside the car, keep the car sealed and stay calm: doors and windows stay shut, and a sealed car means she can be recovered. She has a small, finite space, she will eventually stop scrambling, and she cannot be run over or lost while the car stays shut. Resist the instinct to open a door to reach her or to lunge across the seats, because a grab converts panic into flight and flight aims at any opening. Speak quietly, let her find a footwell or the space behind a seat, and wait until her breathing slows.
When she has settled, create a target: her carrier, opened, with a blanket that smells of home and a little food inside, placed within her reach. Many cats walk into the carrier themselves once it becomes the quietest hiding place in the car. If she must be lifted, use a towel over her and move slowly, supporting her whole body. Check the carrier latch and any damage before the return journey, and wedge or strap the carrier so a repeat performance cannot happen on the drive home.
At the Car Park: No Chasing
At the vet car park the rules reverse. If she bolts as the carrier opens, do not chase her. A pursued cat runs further, crosses roads without looking and hides somewhere far less findable, while a cat who is left alone usually stops within a short distance and hides close by. Note exactly where she went, then set the carrier down near that spot with familiar scent and food inside, its door open, and step well back.
Keep the area quiet. Ask bystanders and the practice staff to keep their distance rather than forming a helpful crowd, and keep dogs away from the spot entirely. Then wait, watching from your car if you can, because many cats creep back to the carrier once the car park empties and voices drop. If she has not returned as light fades, leave the carrier with the practice if they agree, with your number, and return at quieter times rather than searching noisily through the night.
If She Slips Away Completely
If she slips away beyond the car park, widen the net the same day: tell the practice, search the immediate streets and gardens calmly at quiet hours, and leaflet the streets around the practice, because that neighbourhood, not your home street, is her search area. Update anyone who might be contacted about her with the practice's address as the point she was lost from.
For the wider search, Lost Pet Alert System via KinsLe reaches owners living nearby at once, while anyone who finds her can get a message to you without delay.