Very near to you, but not terribly obvious, is a track which runs just inside the trees lining the verge of the eastern side of the Andover Road. It runs for a mile, or a mile and a half and terminates at the John Moore(?) barracks.
It is very slightly downhill, tight wooded singletrack and is a blast if you give it everything in the big ring.
Plan your rides to end on it. There are drove roads that link in at the northen end.
Looks very unpromising on the OS but highly recommended.
Second for Farley Mount and Black Wood, which you would best reach through Micheldever Wood if cycling there.
Happy to meet up sometime if you want company