Fear travels between cats faster than calm does
Cats read each other constantly. Your first cat's yowling in the carrier, the smell of stress she brings home, and the household tension on appointment mornings all teach the second cat a lesson you never intended: the clinic routine is something to dread. The new cat may have no bad experience of her own yet — she is borrowing one. That is actually good news, because borrowed fear can be prevented from becoming her own, provided you stop letting the two cats' vet lives run as one shared, chaotic event and start treating them as separate patients with separate preparation.
Separate their appointments wherever you can
The single most effective step is also the simplest: do not take them together. Book the second cat's visits on different days from the panicking cat's, at least while her habits are forming, so her entire experience — carrier, car, waiting room, handling, return — happens in a calm household and a calm car. Yes, it means two trips. But each peaceful solo visit lays down an association the next visit builds on, whereas one shared trip beside a screaming housemate can undo months. If the clinic offers quieter appointment slots, take them for both cats; a calmer waiting room helps every feline patient in it.
Build her own carrier and car routine from zero
Do not assume she will learn the carrier by watching the other cat avoid it. Run her own programme: carrier out permanently in a room she likes, her bedding inside, meals fed beside and then in it, the door closed for brief, treat-ended sits that lead nowhere. Then graduate to the car — sit in the stationary car together, then a drive around the block that ends back home with dinner, so journeys stop predicting the clinic every single time. Keep these sessions hers alone; the other cat should not be shut in the same room rehearsing her own dread within earshot.
If they must travel together, manage the shared day tightly
Sometimes one trip for both is unavoidable — a house move, a single annual slot. On that day, damage-limitation is the goal. Use two separate carriers, each with that cat's own bedding, both covered so neither can stare at the other, and position them so the calmer cat is not directly beside the panicking one in the car. Bring the calmer cat in first if the clinic allows staggered arrivals, or ask to wait in the car with her until called. Back home, let each cat out in a separate room and give the returning patients time apart, since a cat smelling of the clinic can get a hostile reception from the one who stayed — or the one who travelled with her.
Keep working on the first cat too, on her own track
Separating their routines buys the new cat a fair start, but the original problem remains: a housemate whose panic fills the home on vet days. She deserves her own slower programme — the same carrier-familiarity work, done at her pace, plus a conversation with the clinic about how she presents and what they can do at their end, from first-slot appointments to taking her straight into a room. Tell the vet honestly that her fear is affecting the other cat; it is useful clinical context, not an admission of failure. Coordinating both cats' separate practice sessions through KinsLe Care Teams keeps the preparation fair — each cat's sessions assigned, done, and marked complete — instead of the confident new cat getting all the training while the frightened one gets none.