These poor animals don't like going to the vet, just as much as you don't like going to the dentist. But they have a key rule which seems to help them through the traumatic event... if you don't see them, they don't see you.
It's been proven to work for cats and dogs for many, many years. And it's so easy to do it! First step, when you finally get out of that carrier, survey the area and find possibly hide-out.
Examples of good hide-outs: the sink, in folder holder, behind items, inside a hole of some sort. The hole option is the best option but not always available.
Examples of bad hiding spots: sitting in a corner with your face towards the wall, sitting in an open space with your face against the floor.
However, sometimes, that's just the way it's gotta be!