Hampshire is New Forest pony country, and the New Forest is one of the most genuinely family-suited natives there is — the right size for a child to progress on, hardy, and with generations of breeding for a sensible temperament. Forest-bred ponies are used to traffic, dogs, walkers and everything else a busy landscape throws at them, which is exactly the exposure a nervous family wants.
Beyond the Forest, Hampshire has an active Pony Club and riding club scene and good access to the south-east show circuit. Prices sit at south-east levels. If you are looking at a Forest-bred pony, ask carefully about handling and schooling — running on the Forest produces a level-headed pony but not necessarily a schooled one, and there is a real difference between a pony that is calm and a pony that is trained.