Yes, there is a way out that works without TNT. You'll end up at the train stop for Rail Crossing. This exit is to the southwest of Dude's cave home.
It may be a little tricky to get there without getting killed. I think you want to go mostly west and then south, but I'm pretty sure going immediately west from Dude's cave home leads to a cave that is very likely to get you killed. You don't want that one. Don't get eaten by crawlers, man.
If you find a ford across a river that is guarded by psi beetles you're on the right track, just get over on the west side of the river and go a couple zones south and you're out. There's some interesting stuff on the west side of that river that you'll want to see and may come back to again.
Actually, I think there's another way out, but I never go that way. I think if you go NORTH from Dude's cave you'll (immediately? eventually?) find a patch of rocks that you can climb up into the tunnel system, and from there you could find your way back to Core City or Foundry eventually.
Either way could get you hopelessly lost or killed, but I suspect looking for the Rail Crossing exit is probably easier.
