As already stated, there are lots and lots of lovely walks there.
One of my favourites is to start from Cadnams Pool (grid ref SU 230122) and head into the trees opposite the pond (in a northerly or north-westerly direction) and go steadily downhill until you reach a delightful forest stream.
Be very careful if you cross the stream - there are some very boggy bits - I once plunged in above the knee, fortunately my other leg was still on firm ground and I successfully extracted myself after a bit of a struggle.
I also got a bit lost here once - on a day of very weak sun I emerged into a clearing and realised from the sun I was heading in a totally different direction from the one I thought I was. I then used a compass to head south knowing I would at some stage hit the road, which I did some half a mile east of where I expected.
If you leave the beaten tracks, then map and compass are a good idea.