Small metal structures sustaining plasmon resonances in the optical regime are of great interest due to their large scattering cross sections and ability to concentrate light to subwavelength volumes. In this paper, we study the dipolar plasmon resonances of optical antennas with a constant volume and a sinusoidal modulation in width. We experimentally show that by changing the phase of the width-modulation, with a small 10 nm modulation amplitude, the resonance shifts over 160 nm. Using simulations we show how this simple design can create resonance shifts greater than 600 nm. The versatility of this design is further shown by creating asymmetric structures with two different modulation amplitudes, which we experimentally and numerically show to give rise to two resonances. Our results on both the symmetric and asymmetric antennas show the capability to control the localization of the fields outside the antenna, while still maintaining the freedom to change the antenna resonance wavelength. The antenna design we tested combines a large spectral tunability with a small footprint: all the antenna dimensions are factor 7 to 13 smaller than the wavelength, and hold potential as a design element in meta-surfaces for beam shaping.