Joist strength

I'm starting a project to turn unfinished space over a garage into a closet. The joists over the garage (unfinished space) are 2x6 with a span of 14 feet, 16 inch oc. The second floor joists under the finished areas are 2x12's. My question is: Can I put another 2x6 on top of the existing 2x6 joists with side bracing to achieve the proper strength for the span? Or, should I replace/sister all of the 2x6 joists with 2x12's?

Getting 14' 2x12's up there will be difficult so suggestions/ideas are very welcome.

Thanks to anyone that can help me out.
Jim