@Sara007 - It sounds like your daughter may be using the itchiness as an excuse to keep her head uncovered. My kid used to do this too, even when I purchased something that was silk lined and definitely wouldn't cause discomfort.
What I did to get my daughter in hats was take her to a store that sold all sorts of those ultra cute animal head hats. She loves bears and cats so I was able to get her into an adorable hat with ears and a little kitty face. She loves that hat and doesn't complain the least about it being itchy. I think you may need to go for the fashion angle to keep your kid warm.