I think you'd need some differentiation - if there's a Jones Magazine of library science, you can't have a Jones Magazine of fiction. Especially if they've trademarked the name as then they'll be within their rights to sue you out of existence.
However, you probably could have the Jones Fiction Magazine or something like that without too much trouble, unless they have very trigger-happy lawyers.
If I was in that position, I would probably write the editors of the existing Jones Magazine and ask for permission to use the name for your purposes. If they say yes, problem solved. If they say no, you probably dodged a bullet - you'd be much better off finding a new name early on than waiting until you magazine is successful (at which point you become an attractive lawsuit target) and then have to deal with transitioning to a new title with an active readership.