You don't want to drive with a rusted-through exhaust pipe dragging noisily on the road. Before you get to the muffler shop, here's what to do with an exhaust pipe that is about to fall off.
Wait for the engine to cool down so the exhaust pipe is cool enough to touch. Place your hand near the pipe to gauge how hot it is before actually touching it, or wrap a thick rag around your hand.
If part of the pipe is about to fall off, remove the hanging part by bending and twisting it until it breaks free. You needn't reattach it, since you must have it replaced or welded on at a muffler shop anyway.
If you can't break off the hanging pipe, untwist a wire coat hanger or use some mechanic's wire (available at an auto-parts store). Wrap one end around the hanging pipe and fasten the other end to the underside of the car. Do not wrap the wire around the drive shaft or anywhere where it risks becoming tangled with the drive shaft. Wrap it around a stationary part underneath the car.
Make an appointment at the muffler shop.
Fix-it tape for exhaust systems can only cover up a hole or a crack in an exhaust pipe. It will not hold a broken pipe together.