Rincewind is not shot off the edge in the Potent Voyager, but on it:

He also saw a gang of men climbing desperately over the lower slopes of the launching ramp and manhandling a large baulk of timber on to the track, in a frantic attempt to derail the ship before it vanished over the Edge. The wheels slammed into it, but the only effect was to make the ship rock, Twoflower to lose his grip on the ladder and fall into the cabin, and the hatch to slam down with the horrible sound of a dozen fiddly little catches snapping into place. Rincewind dived forward and scrabbled at them, whimpering.

At this point he is clinging desperately to the top of the Voyager. A few paragraphs later it goes over the edge, with Rincewind still on the outside:

Rincewind stood up. There was only one thing to do now, and he did it. He panicked blindly, just as the ship's bogeys hit the little upgrade and flung it sparkling like a salmon, into the sky and over the Edge.

He then wakes up draped over a tree growing out of the Rim itself:

He was alive and lying covered in bruises in a small thorn tree that was growing in a crevice in a rock that projected out of the foaming white wall that was the Rimfall.

It's not explicitly stated, but presumably he fell off the top of the Voyager as it went over the edge and landed in the trees of the Rimfall rather than falling with the Voyager.

The book ends with him letting go and falling into the universe; in the sequel, as mentioned by @Richard, the Great Spells contrive to get him back onto the Disc.

(The Luggage also goes off the Edge outside of the Potent Voyager, by the way; only Twoflower and the sea troll, Tethys, are inside it.)

Rincewind is not shot off the edge in the Potent Voyager, but on it:

He also saw a gang of men climbing desperately over the lower slopes of the launching ramp and manhandling a large baulk of timber on to the track, in a frantic attempt to derail the ship before it vanished over the Edge. The wheels slammed into it, but the only effect was to make the ship rock, Twoflower to lose his grip on the ladder and fall into the cabin, and the hatch to slam down with the horrible sound of a dozen fiddly little catches snapping into place. Rincewind dived forward and scrabbled at them, whimpering.

At this point he is clinging desperately to the top of the Voyager. A few paragraphs later it goes over the edge, with Rincewind still on the outside:

Rincewind stood up. There was only one thing to do now, and he did it. He panicked blindly, just as the ship's bogeys hit the little upgrade and flung it sparkling like a salmon, into the sky and over the Edge.

He then wakes up draped over a tree growing out of the Rim itself:

He was alive and lying covered in bruises in a small thorn tree that was growing in a crevice in a rock that projected out of the foaming white wall that was the Rimfall.

It's not explicitly stated, but presumably he fell off the top of the Voyager as it went over the edge and landed in the trees of the Rimfall rather than falling with the Voyager.

The book ends with him letting go and falling into the universe; in the sequel, as mentioned by @Richard, the Great Spells contrive to get him back onto the Disc.

(The Luggage also goes off the Edge outside of the Potent Voyager, by the way; only Twoflower and the sea troll, Tethys, are inside it.)