A loooooong time ago, I bought a sweater. It was the prettiest purple. I wore it. I washed it. That was the end of wearing the pretty purple sweater. I had never had a truly un-washable article of clothing before. The purple sweater became my very young daughter's sweater. Then I washed it again... it could not possibly shrink more if it had already shrunk in laughable proportions, right? It become a toddler sweater, except it was too tight (or felted from washing) to actual be worn.
The pretty purple sweater now resides in a bin where I store treasured used clothing to refashion into doll clothing. A thrice shrunk sweater is delightful for making all sorts of things for dolls - hair, socks, leggings, and today - a scarf.
Simply cut the length of the sweater about 1" wide, cut fringes into the ends and wrapped around your doll. Done!