From the Good Human site:
Non-profits like Goodwill and the Salvation Army play a crucial role in keeping old clothes out of the waste stream. When they get donations of clothes that are too threadbare to re-sell in one of their shops, they send them to “rag sorters” that specialize in recycling pieces of fabric large and small. Says Hawley, these textile recyclers sell about half the clothing they get back overseas in developing countries, while unusable garments, especially cotton t-shirts, are turned into wiping and polishing cloths used by a variety of industries and sold to consumers.
I know The Purple Heart, the veterans charity, accepts clothing and sells it to sorters too.