I have no idea how to do what you are suggesting to check, guess I'll need to find somebody who knows what they are doing.
I went back and re-read your original question and I see that you are wanting to utilize the attic for storage only. For some reason I thought your home was almost as old as you are. I'm not sure if Texas had adopted the ICC codes for residential construction when your home was built but that probably doesn't matter. Most building codes utilized the same span charts that are inside the International Building Code. Span Tables for Joists and Rafters is a chart developed by the American Wood Council that is a rule of thumb guide for builders, component manufacturers and inspectors. The information provided in these tables allows a builder to conventionally frame a home in the conventional manner, especially as it relates to the floor, roof and ceiling assemblies typically without the need to have a licensed structural engineer stamp and seal the plans (although a lot of states are now making that a requirement).
Access to your attic-It sounds as though you only have one access point to your attic, Even so, you should be able to reach the garage attic space through the scuttle hole or pull down stairs located inside your home, although it may mean you have to walk across the ceiling joists to get there. All building codes in the last 50 years have had a requirement that would allow access to any concealed space over 14" in height.
I am going to assume your foundation is monolithic concrete slab common to Texas. If the rafters and ceiling joists are constructed with prefabricated trussed components, they are generally engineered and fabricated of #2 - 2x4 Southern Yellow Pine (SYP) with panel points engineered to withstand loading as well as uplift. These panel points are connected with galvanized steel plate connectors. Trusses are normally spaced 24" on center. ( Gren's attachment illustrates a truss)
Two considerations if this assembly is a trussed design.
1. You cannot cut any member of this assembly as it works and depends on all parts as a system as designed
2. Try not to put your storage area in a mid-span area, rather try to locate it near or over a wall below where the truss member has some additional direct bearing. The bottom chord is designed as an integral tie beam and not a floor joist.
If the roof assembly is conventionally framed (not trussed), the rafters are probably 2x6 and the ceiling joists are probably 2x8 and are spaced 16" apart. While it's not ideal to cut any of these members, this system is certainly more forgiving and could be repaired if necessary without an engineer's repair certification. If you decide to add a folding stair in this system, you could cut one joist and add a header and trimmers to accomodate a 22 1/2" wide folding stair. This system is also easier to lay flooring on for either a walkway or storage. I hope this helps.
*edit-be careful not to lay any plywood on top of any Romex wiring or condensate drains