I have small hands but love the tone of 34" scale basses. I use or have used over the years Rickenbacker 4001's, Fender J's and Musicman Sterling. All of the above have been 4 string basses. They all have narrow necks, but are all long scale.
I have to slide arround the fingerboard a bit from time to time but this has become part of 'my' sound.
With your budget the Fender J fits your pocket and hands well.
I have to slide arround the fingerboard a bit from time to time but this has become part of 'my' sound.
With your budget the Fender J fits your pocket and hands well.