From Aldermaston there's a few alternatives…
Head SW towards Watership Down, Hannington – lots of nice bridleways down that way.
Go north of the A4 around Chapel Row – again, loads of stuff.
For general hacking around exploring there's Pamber Forest nearby as SilentSparky says – just east of Tadley – open access trails and just go where your wheel points. Greenham Common has a few bits of interest around it too.
There's quite a few MTB clubs in the area – ask around, or pop into Banjo Cycles in Newbury – IIRC they have organised rides round that way.