Although not found in encyclopedias and dictionaries, a toboggan may refer to a stocking hat as well. From some random sampling at LeTourneau University, I've found that people who know that a toboggan is a sled do not refer to a stocking hat as a toboggan. On the other hand, people who do not know that the original meaning refers to a big sled often call a stocking hat a toboggan (or a beanie). I imagine the rest of the world (i.e. not-USA) call stocking hats by a completely different term.

Since I haven't been able to find much information about how a hat started to be referred to as a toboggan, I assume that people often wore one of these types of hats while sledding on a toboggan (sled). Somehow the term got confused between a big sled and a hat.

To*bog"gan (?), n. [Corruption of American Indian odabagan a sled.]

A kind of sledge made of pliable board, turned up at one or both ends, used for coasting down hills or prepared inclined planes; also, a sleigh or sledge, to be drawn by dogs, or by hand, over soft and deep snow.

© Webster 1913.

To*bog"gan (?), v. i. [imp. & p. p. Tobogganed (?); p. pr. & vb. n. Tobogganing.]

To slide down hill over the snow or ice on a toboggan.

Barilett.

© Webster 1913.