I, too, remember garters!
In the Table scene in episode 1, we briefly see Claire’s metal clasp garters, probably attached to a garter belt, as she raises her skirt. I recall those all too well.
In scenes in the eighteenth century, we see Claire’s tied garters that seem to be integral to the stockings. I wore knee-highs as part of my high school uniform, and we usually used rubber bands as garters, folding the tops of the socks over the rubber bands to hide them.
A couple comments from the books: in a couple scenes Claire mentions men’s garters, so men apparently work garters to hold up their stockings, too. And in one of the books Jamie and Young Ian are shocked that Claire doesn’t know how to knit stockings, a skill both of them have!