How close to Flag?
How far do you like to ride?
Camp overnight or day trip?
It has been 20 years since I lived there and Coconino Forest shut down a lot of single track, but some may be coming back.
There are interesting places to explore east of the road to Walnut Canyon.
Also, off Lake Mary Road on the south side of Walnut Canyon, such as FR 128
The ride down Lake Mary Road to Happy Jack to highway 87, through Strawberry and Pine to Payson is an enjoyable paved ride.
If you want to go on to Payson (highway 260) and even Alpine, you can go over 150 miles without leaving the largest Ponderosa Pine forest in the world.
Along the route are many forest roads