Story arc: The Sims 2 (both of them)
After The Urbz, Giuseppi's storyline is essentially over. You might as well stop playing here, but if you just want one last sip of this series out of the bottom of the cup (and one last bit of Giuseppi's character), there's The Sims 2 for GBA and for DS. And to be clear, they are not the same game, but they share a similar setting. The way they're set up sort of exclude each other - it's a branching timeline and some of it is declared in-game to be non-canon. Yeah, it's weird.
The Sims 2 (GBA) is a game revolving around a reality show that is being filmed with the residents of Strangetown as unknowing actors, one of them being Giuseppi. Maybe he's in the know, but he doesn't tell us. Maybe he's a robot, but that's unclear. Yeah, it's weird.
The Sims 2 (DS), instead, is completely unlike any other game in this series. It is centered around the player character managing a hotel in Strangetown as everyone, including the player, gradually loses their minds. Yeah, it's weird.
Giuseppi is clearly spotlit in these games. Evidently, the developers knew he was a fan favorite. The bonus episode of The Sims 2 (GBA) even flat-out calls Giuseppi one of the favorite characters. But he doesn't do much in either game, and, in fact, is absent more often than not.
What you can take from both of them is that, about one year after the events of The Urbz, Giuseppi happens to be in Strangetown, for one reason or another. What both of these games add to his character is some insight into his childhood and the detail that he's still fond of toys - in The Sims 2 (GBA), he's plays with a paddle ball attempting to break his childhood record, and in The Sims 2 (DS), he sells you a a piece -- he still possesses (and needs to be persuaded to part with) his childhood super soaker. But that's pretty much all there the new information there is about him.
His in-game description in both games state that he's no longer a criminal, just a shady guy - so we can assume that, for whatever reason he's in Strangetown, he's continuing to remake his life in legality. However, he still must be having trouble with that. In The Sims 2 (GBA), he's doing a task for known mafia lord Frankie Fusilli - and while the task is essentially babysitting, it still seems off that Giuseppi would knowingly and willingly work for a mafia lord. Maybe it wasn't quite so willing. Giuseppi's past also seems to be haunting him in The Sims 2 (DS), since, when he stays in the hotel, he mentions that he's there because he needs to lay low for a few days. However, while the area around the Strangetown Hotel is rife with its own crime problems, Giuseppi is notably not involved in any of them.
Based on leftover code present in The Sims 2 (GBA), it seems like the developers had ideas for a lot more in-game episodes, which, for some reason, didn't get made. One of them would have featured Giuseppi being framed for a crime he claims he didn't commit, and the player would need to unravel the mystery of what really happened. That's all we know about it. What a shame this didn't make it into the game. That would have given Giuseppi more of a presence in The Sims 2 (GBA), and it would have given an opportunity to show more important aspects of his character than just having him shovel manure (really).
Another scrapped episode was apparently called "Jumping the Shark", as a parody of that moment in Happy Days when the fans all knew the writers were out of ideas. It was Fonzie who did the literal shark-jump in Happy Days, and Fonzie is clearly an inspiration for Giuseppi. Eyyy! However, as we can see in a leftover cutscene, it would have been Dusty Hogg to jump the shark - which makes sense, since he's the character with the motorcycle.
Anyway, the way we see Giuseppi in these two games is probably indicative of what will be the rest of his life: he's traveling everywhere, not being a criminal, but being seen as a shady guy. But still not giving up the leather jacket, and always up for a friendly Eyyy!
We hope some day he gets another van, and can feel free.