** BUT! Why are towels and shirts getting into the vent? **
I assume the OP is adding such cloth to the snake to make it a larger diameter to fill the pipe to completely clean the vent line? There are however tools designed for dryer vent cleaning. One can be seen at the following link:
I don't know if one would be strong enough to get out a piece of clothing wedged into a dryer's vent though, I kinda doubt it.
I can't think of any other way to get it out except pushing it right through to the other side. That may also make sure the vent is really clear this time? :)