I'm afraid I can't help with suggestions for places to live in Hampshire, but just to add to the general background information about the county and the use of the name, you'll find that Hampshire is often abbreviated to "Hants." in postal addresses, which might seem rather odd at first.
The origin is historical and there are two conflicting theories. One is that is derives from "Hantshaving," another name for Hampshire which appears to have been used in the 19th century.
The other is that it's from one of the Anglo-Saxon variations of Hantum + Scir -> Hamtunschire -> Hantescire (Hantum being what is now the city of Southampton and Hantumscir et al meaning "the area governed from Hantum" -- the origin of the modern -shire endings of our county names).