One idea would be to take/cut the bottom bracket off the existing frame and make extensions. The extension I'm envisioning would have 2 pieces. The first piece is the main, "outer" piece and is your desired 6" long piece and it matches the outer and inner diameters of the existing frame. If it was mounted using set-screws, the original bracket can be reused and put at the bottom of this first piece. The second piece of the extension is an inner tube whose outer diameter is the same as (actually just a HAIR smaller than) the inner diameter of your existing frame's tube. This one would be cut to be about 12" long. The inner tube slides inside the main, outer extension tube and also into the existing frame. Set-screws, pins, or bolts are used to attach the inner piece/tube to both the existing frame and the outer piece of the extension.
Another idea would be to use something like this:
Stainless Steel Rail Fitting Tees 90 deg 1-1/4"
Stainless Steel Connection 1"
I'm not sure how well something as short as those would work if someone pulls on/leans on/falls against the bimini, though. I think I'd prefer to have some of the force distributed along a bigger portion of the frame, that's why I suggested the set-up above. Of course, never having actually done any of these options, I could be WAY off base, but...
And, of course, the most obvious way to get the desired 6" is to buy a new bimini that is taller than you need and "just" cut it down.