What other antibiotics could be used to treat a patient diagnosed of acute or chronic rhinosinusitis when augmentin (amoxicillin and clavulanate) did not work?

Many: Different bacteria live in different regions of the world. Certainly, the species of bacteria are very similar at their root, but many bacteria of the same species have developed different resistance to different medications around the world. A fluoroquinolone may be indicated in this case, but without knowing more about you or your symptoms, it's hard to say. Go back to your doctor.