I've flown in and out of Southampton these past thirty years. Southampton's problem is space - lack of it ! Others have commented similarly. Bounded on the south by the M27, on the west by the M3 and sundry secondary road routes and on the north by the railway track, buildings and assorted sidings and as many chimney pots north and south as you won't be able to count.
The answer to an effective regional airport located virtually in Southampton lies in development of Lee-on- Solent or, if you prefer, the former HMS Daedalus. The last time I looked it was a haven of aviation surrounded by green fields on the edge of the Solent. There would be transportation/connection problems to overcome but, it looks, at least superficially, as worthy of a feasibility study. Indeed, it might have been done.