I use old hand towels and dishcloths. You can also use old washcloths. If you don't have old ones go buy some new ones and then you'll have some olds ones.
BTW: the towek you wrecked is now perfect for the job.
From time to time throw out the nastiest of your towels and keep the supply of new old towels moving.
I tend to keep different levels of dirty towels and they work their way down the chain until they are no longer useful.
They start their lives cleaning the frame; then they move down the ladder and clean the wheels; then the chainstays; then the derailers; then the chain.