3/4 hardwood install in 23 x24 room

We have just completed an expansion of our great room from 14 x 16 to 23 x 24 and are going to install 4" wide by 3/4 thick plank flooring. I know from the previous thread and other literature that you should be installing the floor perpendicular to the joists. The problem is the joists of the expansion run opposite to the joists of the original room. The subfloor is 3/4 t&g. If we add 1/2 on top of this for rigidity it will put us out of whack with our current adjacent tiled rooms. Is blocking between the joists a remedy (if so, what is the minimum size and spacing that we can get away with). If this is not really an issue let me know that also.

Much appreciated. Bill T