Yes. It can spread to your eyes or the genitals. This could happen if you rub your eyes after getting saliva from an infected person on your hands, or if you receive oral sex from someone who has cold sores.
Don't pick at a cold sore. You can spread the virus to another part of your body. That will make your outbreak worse. Keep your hands away from your mouth, and wash your hands often, especially when you touch your face.