Cooling Springs has a total of 16 puzzle pieces. There are 4 puzzle pieces in each sub-level. While looking for the puzzle pieces in Cooling Springs, it is a good idea to look for all 9 artifacts as well. This will reduce the amount of time needed to find collectibles and will prevent you from having to back-track later in the game. The video below contains all artifacts and puzzle pieces in Cooling Springs.
Credit to PS5Trophies for the video.