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When I started, I picked the hair I wanted, but for a flirt thing I had to buy a new one. Can I get my old hair back without paying for it again?
No. Unfortunately you will have to pay for it again.
To add more info, that applies only to hair from the 'makeover' section. There are a few 'wigs' you can buy in stores in the accessories section. Not all stores have them. Those work like normal clothing items.