From what I've read above the focus is on re-joisting and that is correct, but in my opinion only partially. My concern would be supporting these ceiling/floor joists not only at mid-span as you have noted, but as importantly at the outer walls. Not knowing if the existing wall studs are 2x4s or 2x6 will have a direct bearing on the support & rigidity of the structure. Considering the potential "living" use of the upper floor of your garage, or for heavy storage, the walls should be constructed of 2x6s set on a good 8" foundation wall on a good footing(8"x16").
If the sill plate on the foundation is 6" and the existing wall is 2x4, then I would consider building another wall inside the existing wall with the new wall studs staggered between the existing studs at 16" with a crosswise stud sitting on the sill plate where the 16" matches an existing 24" stud and then supporting the new 10" joists on top of the new double top plated wall & the mid-span walls. The tops of the new joists would match the tops of the existing 6" joists. This would reduce the inside wall-to-wall dimensions of the garage a little, but would provide the support needed & you would not be concerned about this in the future.
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