Welcome to Just Answer.
Sorry to hear about your problem.
I would use a plumber's snake/ auger which has a corkscrew device on the end and it is good at hooking cloth like objects. These can be bought at your local hardware store/ home improvement store.
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Best of Luck, Brian
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Yes, because it is very flexible; it is wire that is coiled tightly, like a spring, which makes it stiff but flexible. I just used one in a toilet last night and it worked great; it actually will hook the cloth and allow you to pull it back.
Yes, sorry I forgot to tell you that you could go down the overflow pipe, by removing the cover plate, as a second option. You will also need to pull out the lever and linkage. The end picture shows a typical cutaway view of a tub drain.
Does the overflow cover look similar to this? If so, look around the edge for a small set screw that will need to be loosened.
There could be a center screw on this...it is hard to tell from the picture.
It could twist by hand rather than by screwdriver. Does the cover have a ring like the picture below?...if so turn the ring to loosen the bolt.