A detailed characterization of plasmon modes is important not only for a deeper understanding of plasmons but also for their practical applications. In this study, we investigated the three-dimensional near-field characteristics of high-order plasmon modes excited in a gold hexagonal nanoplate. From the near-field spectroscopic images, we found that both in-plane and out-of-plane plasmon modes observed near 900 nm were spectrally and spatially overlapped. We performed three-dimensional near-field measurement to reveal the optical characteristics of the overlapped modes in detail. We found that the steric near-field distribution near the nanoplate strongly depended on the plasmon mode, and the out-of-plane mode confines electromagnetic fields more tightly than the in-plane mode. We also found that the in-plane mode was dominantly visualized as the probe tip-sample distance increased. These findings demonstrate that the three-dimensional near-field technique enables selective visualization of a single plasmon mode even if multiple modes are spatially and spectrally overlapped.