Take the pillowcase off the pillow you slept on last night. Stick the cat in that. Then stick that in the carrier. Your cat will be surrounded by your smell. That is very calming for your cat. I didn't come up with this idea. I read it in a cat book as a kid.
For mine, at the end of the vet visit, I just had to hold up the opening and my cats walked in. In fact, they were usually trying to get back in the pillowcase the entire vet visit.