Here is "Have you got a hobby?" in a book by a British guy (I think Scottish).
Here is "What hobbies does she have?" in a book by a British writer Louise Beech.
Here is "What hobbies has he got?" in a book by an American journalist (from 1969).
However, your question with "has Jessica got" does sound a bit unnatural, and I'm just going to throw out some guesses as to why that might be.
The thing is, "'s got" as a synonym for "have" has become something of an ungrammatical idiom, to the point that it's now OK to ask, "What do you got here?" (meaning "What do you have here?")—although it is, of course, very much a colloquial expression.
Twisting and turning it back and forth into different positions in the sentence trying to adhere to the rules of grammar does seem to blur the lines between "get" as in "obtain something" and "have got" as a synonym "have".
Leave "have got" to simple sentences like "I've got/She's got" or a simple question "Have you got?" and don't shun sentences with "does she have."