In reply to TRJ: any coppice or hedge will do at this time of year, I used to regularly stay in the woods all over the area. Never had any problems with farmers, just occasionally the odd poacher/lost squaddie. There are numerous old green lanes or coppices on the downs, and the scarp slope is often wooded.
Foxhole is a good backpackers 'pay' campsite by cuckmere haven, if you go up via Alfrison theres The Comp green Lane, or the woods below Firle beacon (pheasants too, Ram Inn used to be good before the people of Firle torched a gypsy caravan for Nov 5th), further along gets more wooded into west sussex.
There are standpipes all along the route, or cattle troughs if you've a filter/like the taste. I dont know if its off route but the water in the Meon valley is pure and drinkable (watercress beds). In season you can snaffle some nice greens or root veg, especially in Hampshire (which is more wooded).
Just check your map for a spinney and doss down. Check for old barbed wire before you choose a site, and dont be near badgers! Chanctonbury Ring is good site(also Romano-Celtic temple there, cult of the pig I believe). Nothing like waking upin the frost to the sea on one side, and the Weald in mist on the other!