I beat the game and unlocked all the cars, but I'm wondering if there's any reason to keep playing - ie, if playing with a new car can open up new, hidden levels or something like that.

Back in the day, I used to just run around to get max time on the timer. From memory, it displays 19:59. But it just keep racking up after that. Always leave some peds + don't wreck all the opponents. Then, I just used to explore the maps. Get the car to places you couldn't ordinarily get to, like on the roof of the football stadium in one of the maps. Looking for secrets, that sort of thing.
I don't remember finding any secrets. There probably aren't any. Anyway, there's not really much reason to carry on playing after finishing the races imo, other than exploring the maps fully. Doing crazy stunts, jumps and whatnot. If there are any secrets, they are probably well documented on fansites. It's almost 20 years ago now since I played it to that extent. When I picked it up here, a couple of years ago, I played through the game, and was done with it. Nostalgia is great and all, but no way am I putting that sort of time in to it now.
Holy moly, that was a long winded way of saying "no, not really". You could get in to modding, I suppose. but I think I'll stop here before I bore you... :)

Okay, thanks. :-P I just finished the Splat Pack, and frankly I thought the tracks were MUCH worse than the regular game's. Too few open areas to just smash around.

But at least I've got the game under my belt.

I keep playing because it's fun. Plus the sweet feelings of nostalgia are a rare and intoxicating kind of pleasure.