I used to live in Lymington and found it quite challenging to find good riding in the New Forest – all the stuff you're "meant" to ride is boring, undulating loamy doubletrack footpaths. You need a bit of local knowledge to string together any decent lengths of the good stuff.
Your best bet is to check out Newforce , the New Forest Mountain biking club, and try and tag along with their rides – they do a midweek one and a Sunday one, IIRC.
From memory, the best riding to be had in the Forest was in the area round about Fordingbridge – more lumpy hills and just generally a bit more variety than elsewhere.
The coastal path from Lymington on towards Christchurch through Milton and Barton on Sea and Mudeford also has some nice swoopy bits here and there, although again, it's all a bit cheeky. Makes a nice there-and-back on a summer's evening and is hard to get lost on.