I just had this problem! First vet appointment was today! I got a plastic tote and put a screen lit on it. Then put that in a bag (like one of those reusable grocery bags, large bag purses or whatever) to limit his visibility, so he wouldn't have to be stressed about being lower than me (buckled in the car while I was driving) or the dog in the vet clinic. I kept him with no visibility until we got into the actual room, then I opened it up.
I noticed how COLD their office was!! I had put a few damp paper towels on the bottom, and with only the top being screen, it was keeping the humidity at a nice 70%. My issue was the temperatures! It was already 95 outside, so outside (and obviously in my car without a huge amount of air conditioning - I still needed some to breathe) the temperatures were fine, but the vet clinic was cold cold cold cold cold because they had the air conditioning BLASTED! So even putting one of those little heating packs that you break the thing and it lasts so long might be nice just to put in there for the duration of the actual appointment when he wasn't being handled.
Dont forget a branch (only 1)! It is definitely key as well A) to get him out without having to reach in a hand over him like a predator and B) to keep him off of the paper towels that will be freezing at the end of a full appointment!
Good luck!! Sounds like we were seeing similar things, and it definitely took a huge weight off of my mind when I got him home!