Yes. And her own hat would be il proprio capello.
But this is an exception where you can omit the possessive adjective. See # 2 'omission of possessive adjectives' on this page:
As I pointed out just above, the possessive pronoun is omitted in this sentence. Here is a quote from the link that follows ..
In general, possessives are not used with parts of the body or clothing of the subject when they are the object of the action taken by the subject.
Examples: Mi sono lavato la faccia (I washed my face). Gianni si è tolto il cappello (Gianni took off his hat).
It's wrong because when you use one 'p' it means something different. "Capello" means "hair" and "cappello" means "hat". Oh, and it does make a difference when spoken! You hold on to the sound of double letters longer! In this case, the longer sound means the difference between turning your hair and turning your hat!
Turn in english can also mean swivel or spin. If using turn it might be said she turns her hat around. Like putting a baseball hat on backwards. I think DL could upload a picture of the event and then we would know the type of phase used in english for the action, which ,may not be a direct translation and vv.