I know Chawton Park Wood quite well, and for a period of 10 years or so, would ride in them at least once a week, although since moving from Alton my visits are probably once or twice a month now. Took daftvader round there a month or so ago, good ride it was too :-)
There’s a reasonable amount of single track and cheeky trails, some less obvious than others and since the winter storms, some have some new and as yet indistinct diversions. generally, when you spot a cheeky trail off one of the main paths, it will be worth heading off and following it. You won’t get lost!
If you follow a north-ish direction, you can link up with Bushey Leaze woods, just outside of Alton, by a trail opposite the Jubilee playing fields at their southernmost, , ‘Sleazy Bushes’ are slightly smaller than CPW, but again, quite a few trails to have fun on.