I've done this several times and have devised a good trick. you use one of those flexible extension grabbers:
Tie the end of a 4' length of string to an 8d nail, drop the nail into the carboy and use the grabber tool to thread it through the bunghole, then pull the nail back out using the grabber tool. You now have the string running through the bunghole of the carboy stopper and both ends coming out the neck of the carboy. You may need a little soap or water to lubricate it it through the neck but that's how you get it out.