Hi,
Someone more knowledgable will surely be along soon but I’m not sure you’ll find anything as easy to navigate as Swinley in the area. I live in Fleet and ride Swinley a lot, but there is also a lot of MOD land that can be ridden, however as with the Surrey Hills the trails aren’t that clearly defined. For the most part you need to just ride around, get lost a few times, and get to know an area.
If you prefer XC (not much else round these parts!) you could always try one of the many Gorrick race locations in the area, check their website but off the top of my head Minley Manor, Yateley Common, Tunnel Hill, Fleet/Church Crookham. If they’ve raced recently then so long as you can find the course they should be fairly easy to follow.
There may be course maps on the Gorrick site, or if you use GPS perhaps some kind soul on here has GPX files you could use.
Alternatively why not hook up with either an LBS organised ride or a club like Berks on Bikes or Tunnel Hill Trolls? There must be a club in the Basingstoke area.
Good luck!