Probably won't find a silk one or a child, especially an antique one since it was not general wear for children. Now you may find a small mens hat (under size 5) that may fit on a child but not a baby that is old. Try Ebay for your search, did a quick search under top hot and then narrowed it down to men's clothing and came up with quite a few some beaver some silk some old some new some costume while others dress.
One possibility is to find a company that makes hats and have one or two made that are small-not a one size fits all thing anyway.
http://www.2darksisters.com/gothichats.htm
makes smaller hats-may make even smaller than listed, just email them.