We visualize plasmon mode patterns induced in a single gold nanorod by three-dimensional scanning near-field optical microscopy. From the near-field transmission imaging, we find that 3rd and 4th order plasmon modes are resonantly excited in the nanorod. We perform electromagnetic simulations based on the discrete dipole approximation method under focused Gaussian beam illumination and demonstrate that the observed near-field spectral and spatial features are well reproduced by the simulation. We also reveal from the three-dimensional near-field microscopy that the 4th order plasmon mode confines optical fields more tightly compared with the 3rd order mode. This result indicates that the even-order plasmon modes are promising for enhancing the light-matter interactions.