A "stok" or "stocc" was there lower part of a post or tree trunk in middle/old English. It became the name for any close-fitting garment that covered the foot and lower leg (stok of a person). So really it's not that the name went from being one thing to being another, it's that both the things you mention fit the definition.

The other part of the answer to your question is the development of knitting machines able to create stockings from fine cotton or silk thread made the second type of thing more attainable.