Chilbolton is a very friendly microlight club, and would be quite happy with short-field light aeroplanes, but (a) 410m runway, and (b) no club or rental aeroplanes, and I don't think any syndicates at the moment. it's basically somewhere to take your own aeroplane.
Popham is a wonderful place to operate from, affordable, nice little clubhouse, loads of syndicates, two good runways, friendly people. Irv Lee, already mentioned, operates as an FI from there and is always the best source of local wisdom.
Thruxton is a school airfield, and I've always found it good for that, but somewhat sub-optimal for most post-qualification private flying.
Lee on Solent has transformed in recent years, and become a really friendly place to operate from. Not sure what's there in the way of syndicates and rentals, but I go in there regularly for lunch on the beach.
Old Sarum is another great friendly place, a few syndicates, a busy healthy school, a few local manageable "issues" about the co-located parachute school and proximity of Boscombe Down. Single grass runway, but perfectly adequate for most purposes.
Having flown around that area for many years, I would start at Popham as my first choice, then check if anything else looks at least as good.
G