Unfortunately, it is difficult to get away with much in the New Forest. As you say, it is possible to stick up a tarp & hammock or a small tent after dark and be on your way by first light, but that is probably not the most relaxing experience. You would still have to watch out for the myriads of ponies who blunder through the woods at all times - you really don't want them trampling over your tent while you sleep! The forest is covered by so many laws and regulations that bushcrafting as we know it is all but impossible legally. For instance, there is no wild camping allowed, fires are not permitted anywhere, you cannot take any wood, dead or green, nothing may be attached to trees (not even a tarp line) etc.
However, there are a few remoter patches where you are less likely to be bothered by walkers / cyclists / dog walkers / rangers / keepers / agisters and, in practice, no-one will notice if you were to help yourself to an occasional bit of birch but it is certainly not the wilderness it sometimes appears to be from the outside (unfortunately!).
There are loads of legal camp sites though, ranging from the enormous 3000+ pitch variety, such as Roundhill, run by Camping in the Forest to small private ones such as Acres Down Farm. The forest is just starting to show its autumn colours now and is beautiful as ever.
If you do come down, I hope you have a great time.