Honestly because of vet checks, I've normally never really quarantined a horse; The only place I've ever really had the ability to do so, was at home, when I was still living with my parents, but we had two very large pastures, so normally we would just pen the new horse near enough to the other horses so they could get used to each other, and then just turned the new one out after a few days.
When I got my mare last year, she was in her own pen, but the other two horses and her could touch noses, sniff, etc, over the fence. My girl is a very easy keeper, so I didn't want her hogging all the hay from the two younger horses, and turning into a beached whale! Haha.
I am guessing the combination of the chilly weather plus the travel just caused a cold in your new horse; If you want to help her cough a bit, try Cough Free; it's an herbal remedy for heaves, colds, etc, for horses. I always have that stuff on hand, 'just in case'.
"The ideal horseman has the courage of a lion, the patience of a saint, and the hands of a woman..."