I just found out the reason why meg is wearing a hat when I watched an episode from the official channel of family guy. The story went when Meg's boyfriend is telling her that he's gay. He says "Meg, please stop. I know your heart is in the right place." That scene reveals why Meg always wears a hat. She told him the her heart isn't in the right place and it's on her head.
And also, the doctor said that she should always wear a hat so that's why.
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