Excitation of surface resonance modes and presence of resonance-free hyperbolic modes are two common ways to enhance the near-field radiative energy transport, which can find wide applications in noncontact thermal management and energy harvesting. Here, we identify another way to achieve the super-Planckian thermal radiation via hyperbolic surface phonon polaritons (HSPhPs). Based on the fluctuation-dissipation theory, the near-field radiative heat flux between bulk hexagonal boron nitride (hBN) planes with the optical axis perpendicular to the radiative energy flow can be 120 times as large as the blackbody limit for a gap distance of 20 nm. When the film thickness is reduced to 10 nm, the radiative heat flux is found to increase by 26.3%. The underlying mechanism is attributed to the coupling of Type I HSPhPs inside the anisotropic hBN film, which improves the energy transmission coefficient over a broad wavevector space especially for waves with extremely high wavevectors. This work helps to deepen the understanding of near-field radiation between natural hyperbolic materials, and opens a new route to enhance the near-field thermal radiation.