Doubling the size of a closet with two rods
The blocks of wood your refering to serve two purposes, a resting place for the shelf above them and to provide a sturdy place for you to screw in the holders for the closet rod. 

I understand once you move your rod up and install another one lower you probably won't use a shelf but I still recommend using the blocks of wood. Sure you could screw directly into the drywall, and maybe even hit a stud, but the chance that your closet rod will fall is much less if you use this block of wood.

If you have a hard time getting to a home improvement store to get wood, you can probably even use a board taken from a wooden pallet for this purpose. Just cut it to size, paint it up and it'll be fine.