How to add closet rod support?
Most often the ends of the closet rod are captured in some type of round pocket holders that attach to the shelf support board. So the rod could be removed from these and thus be re-positioned. 

Your easiest solution may very well be to purchase two more of these same support brackets and install one on each end and one in the middle as you showed in your picture. The rod can then be supported by the three new brackets in similar manner. 

I am not a huge fan of metal closet rods and by far prefer a wooden rod. With the three bracket solution you have the option to replace the metal one with a wooden one.