Hale near Fordingbridge? That's very near to me (but I'm a bit stuck this w/e)
Fritham to Frogham is OK, and one of the few permitted rides (but you could do better 😉 )
I've a mate with a satmap GPS & he's got some nice routes that we've done on nightrides. There's a couple that pass very near to where you are. If you want I'll try to tap him up for a route or two (15 miles each(ish), couple of hours). It is a bit of a bog in parts and you do need a map (I can lend you one) or GPS if you don't know the area (just exploring can get you very wet feet – we spent 20min walking across a bog last night, WITH the satmap)
It's generally nice & dry just now though. Supposed to rain tomorrow & brighten again. Unless it chucks it down things'll be nice offroad (quite sandy in much of that area anyway, so better when damp)
If you just want to prat about in slightly hilly woods & can give you some tips too (there's a little jump or two in parts)
ALternatively, there is a nice bit of roadage round there – a niceish 20 mile or so loop using part of the roger penny way just down the road from you, & easily extendable to more
If you can't make the BBB and want more info, let me know