I am intrigued in your post as to why you ask for 'a birthplace not in Hampshire'? It there something that makes you think he was born elsewhere? in the 1841 census, although no place of birth is given, as in later census, there is a column whether headed 'Born in county' 'Y' or 'N' there is an 'N' against his name meaning he was not born in the borough of Southampton, and there is also an 'N' against Caroline WEEKS, I believe this is actually meant to be ARNOLD.
If you PM me I will send you a copy of the birth cert of Caroline and George Bakers son Alfred John and try explain what I think the details should be!!
Ted Beard