We report on an optical fiber surface plasmon resonance sensor that exhibits multiple resonance peaks. The sensor is based on a uniform-waist single-mode tapered fiber coated on one side with a thin metal layer. Owing to the asymmetry of the sensor structure, the different hybrid surface plasmon modes supported by the semicircular layer can be excited by the fundamental fiber mode. As a result, the sensor transmission spectrum exhibits several dips that depend on the taper waist diameter. The advantages of a plasmon resonance sensor with multiple dips are discussed.