một is not an article, the translation of the indefinite a. you use it when you want to quantify an item as one. you won't use it when the number is not important. here, the information that really matters is just hat, the concept of hat, any hat. who cares about the number, there's none anyway. when you say "có ấy không đội một cái mũ", you can imply that she does not wear just one but many hats. if you insist on using một, you should then say "không đội một cái mũ nào", "không đội một cái mũ nào cả", "không đội cái mũ nào hết": doesn't wear any hat (at all).
as a final note, VNmese usually leave out "một" in negative sentences, but to be grammatically correct, an English countable noun needs an article when singular or should have its plural form.