We regularly camp at either Longmeadow (new park farm) or Aldridge Hill. Both are near Brockenhurst and have miles of good gravel trails straight out of the site.
Longmeadow is essentially a flat field with big pitches and plenty of space. Good facilities- shower block and often a food van in the evenings. A few electric hookups. No firepits but can have BBQs off the ground. Site is only a couple of hundred metres from Boulderford Bridge and a nice river for paddling.
https://www.longmeadowcampsite.com/ Deer regularly roam through the site from the adjacent fields.
Aldridge is not open that often and is a very natural site on the edge of a plain, with a small river running through (good for paddling). The site is not sloping, but is bumpy and really best suited to small tents or caravans/motorhomes. There are no facilities (except water and waste disposal) here so will need to provide your own chemical loo.
https://www.campinginthenewforest.com/campsites/aldridge-hill/ Hollands Wood campsite is just across the road from Longmeadow. It's a nice site, but is rather in the woods.