Although I haven't taken a class on GR yet (I'm about to start my second year of undergrad), I'm familiar with the broad concept of how wormholes exist as a certain solution to Einstein's equations, albeit a solution that we probably can't build because they need negative energy.

What I'm wondering is, does anything prevent a wormhole from linking a region of normal spacetime to a region of spacetime within the event horizon but away from the singularity of a black hole? Could such a worm hole allow an infalling obeject to escape? Or would the other side of the wormhole become a black hole, since the gravity of the black hole could be exerted through it?