You don't need more metal cross pieces, looks like you could fix/improve the floor strength greatly with just a table saw and some common dimension lumber.
I'd do like you said earlier, measure how high it is from one of those nice bottom flanges on the cross metal beams, to flush with its top, where the flooring wood is screwed to.
Set the table saw fence and rip some common 2x whatever lumber into strips that wide. Now measure the length between one metal beam edge to the next facing metal beam edge, and cut two of your new wood supports that long. (sand the lower edges where the piece will go into the bend of the metal beam, so that it fits nicely)
Place the newly made pieces where they'll divide those big spans into thirds, or better. Once they are all laid in there put the floor wood back in place and screw it to the metal as it was, and screw it with 1 1/2" or so wood screws, to each of your new supports.
Done. Cheap. Strong.
Edit: Just checked under mine and it's flanged on one side only. Still some edge-on runners screwed to the underside of the floor would stiffen the heck out of the floor.
I'd do what you said already.
