How do I remove broken bathtub overflow drain?

My bathtub is clogged. I removed the bathtub overflow drain to try to unclog it. To my surprise, the overflow drain have a broken chain at the end. This leads me to suspect that the reason for the clog is the "stopper" (sorry, don't know the name for it) breaking off from the overflow drain and getting stuck in the pipe. Is there a shortcut that I can try to grab the stopper? I'm assuming that, worst case, I'll or a plumber will need to remove the piping, but I'm hoping to avoid that.