A "tire" is a headdress, and I think that because Hero is about to be married, it would be a fancy one, such as the net of gold and pearls shown in the image of Catherine de' Medici.
The hair mentioned would be human hair or horse hair woven into the "new tire" that is "within" (i.e., in another room). In the portrait below, Queen Elizabeth is wearing such a tire, though the only part of it that is visible is the row of pearls.
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