I don't know the names of any of the highways, so what I say will have to be slightly researched.
You can go up to Payson, there's some mild twisties that way.
You can take Apache Trail (Main Street) up to the lake, (the fun starts at about Ironwood / Idaho roads). Beyond the lake is a dirt road for about 15 miles, so once you're at the lake, turn around.
If you get a newish map, you'll find a shitload of twisties in the northern parts of AZ. Granted, this isn't the easiest thing to do on a day's notice, so you may want to make a weekend out of it, and have a big twistie-fest.
On a day's notice, though.. Go up to Payson, or go to the lake (the name of the lake I don't know.. I can get there, but to put it into words.. I'm fucked)
Anyway, they're out there, you just have to get a map and hunt them down. It takes more research than it should.