Hi Chap,
Heard that the last off road day at Matchams was cancelled due to some kind of planning breaches, no others are planned.
The other guys are right about New Forest byways; there are only 2 or 3 which if you blink you miss them.
Dorset has 1 or 2 actual byways somewhere, I keep meaning to try to contact the Rights Of Way Officer (if D.C.C. has one) to double check on some other old unmaintained roads that I remember between Sherborne and Dorchester where I grew up.
I go all over Salisbury Plain up to the M4 corridor with other loons and keep finding new places to see and drive, (I will not say that I know it that well yet)
OS Explorer maps seem to have best up to date detail these are the ones I carry which cover a wide area:
Shaftesbury/Cranborne Chase # 118
Salisbury/Stonehenge # 130
Malborough/Devizes # 157
Newbury/Hungerford # 158
Romsey/Andover # 131
Winchester # 132
Basingstoke # 144
Warminster # 143
Not cheap at 8 to 10 quid a time, but it saves you looking at the wrong end of a rifle barrel by going out of bounds.
Hope this helps.
P.S. Should be going on a jolly in a couple of weeks, you welcome to tag along.