It's been previously suggested that old clothes could be used as patches: A sheep wool XsockX plus a spider silk XXsockXX creates a sheep wool xsockx with patches of spider silk, and staves off your Clothier's skill rust for a while. There's no reason why rags shouldn't be a part of this: That spider-silk XXsockXX just becomes a spider-silk rag, which could be used to repair any garment, not just another sock. Rags would have their own wear counters, and Cleaning would become a task done much more often, with craftsdwarves (or any idlers, really,) wiping down their workshops every now & then. (Especially Butcher's Shops--can you imagine?)
Rags could also be used as replacement wound-dressings, after they've been Cleaned with soap. They (or regular cloth, but obviously rags would be cheaper and likely plentiful) should also be a required component in making splints, as the limb must be bound to the wood with something.
Rags would also be used in the manufacture of torches (once the Lighting arc takes effect), and the stuffing for pillows & gambesons, if those ever get implemented, and even making some low-grade paper (depending on what the rags are made of, of course).
Old clothes (and therefore rags) could also be upcycled into stuffed children's toys.
New thought: XclothesX could be given away to fortress visitors, especially those who petition for citizenship but are denied--at least you aren't turning them away with nothing.