Extremely high sensitivity is the long-time pursing goal for surface plasmon resonance (SPR) sensors, which means the capability to detect analytes with lower concentrations or interactions for smaller biomolecules. It was suggested that a flat dispersion for the plasmon material enables to enhance the sensitivity dramatically. However, the regulation ability of inherent dispersion in natural materials is limited. With the appearance of hyperbolic metamaterials (HMMs), the management of the plasmon materials dispersion becomes possible. In this paper, the HMMs with metal/dielectric multilayer structure are proposed to construct the SPR sensors. Through numerical simulations and theoretical analysis, we find that the flat dispersion in the HMM structure could significantly benefit the sensitivity improvement for HMM-SPR sensors. As a result, the proposed HMM-SPR sensor obtains an ultrahigh sensitivity of 34.00 μm/RIU in the experiment, which is one order higher in magnitude than that of the conventional monolayer metallic SPR sensors. Thus, this research shows insight into understanding the dependence of the HMM-SPR sensitivity on dispersion management and provides a practical methodology to develop extremely high-sensitivity SPR sensors based on artificial plasmon materials.