Hyperbolic metamaterials (HMMs) have attracted increasing attentions because of their unique dispersion properties and the flexibility to control the dispersion by changing the components and fractions of the composed materials. In this work, for the first time, we demonstrate a plasmonic sensor based on a side-polished few-mode-fiber coated with a layered of HMM, which is composed of alternating layers of Ag and TiO2. To optimize the sensor performance, the effects of the metal filling fraction (ρ) and the number of bilayers (Nbi) on the HMM dispersion are thoroughly engineered with the effective medium theory and the finite element method. It is found that the HMM with ρ=0.7 and Nbi = 3 can provide the average sensitivity of 5114.3 nm/RIU (RIU: refractive index unit), and the highest sensitivity 9000 nm/RIU in the surrounding refractive index (SRI) ranging from 1.33 to 1.40 RIU. The corresponding figure of merit (FOM) reaches a maximum of 230.8 RIU-1 which is much higher than that of the conventional silver film based SPR sensor. The influence of ρ and Nbi on the sensitivity are well explained from the aspects of the electrical field distribution and the dispersion relationship. This work opens a gate to significantly improve fiber plasmonic sensors performance by engineering the HMM dispersion, which is expected to meet the emergent demand in the biological, medical and clinical applications.