The ability to engineer the optical response of a plasmonic nano-object is highly desired to achieve better control over light-matter interactions. Due to the sensitivity of plasmon resonances to the surrounding media, isotropic dielectric coating is an easy approach to modify the optical properties of a plasmonic nanostructure. However, the choice of coatings and the provided tunability is limited by the range of refractive indices of available materials. Here, we show that coating of plasmonic nano-objects with an anisotropic metamaterial, which displays a hyperbolic dispersion and allows the design of refractive index on demand, provides greater flexibility to engineer their interaction with light. This is experimentally demonstrated by coating Au nanospheres with alternating SiO2 and Au multishells, which creates rich and highly tunable plasmonic modal structures covering a broad wavelength range (~400-2200 nm) and produces high local field intensity enhancement (~500 fold). The concept is extended to hyperbolic coating of dielectric nano-objects, confirming the nature of the modes to be related to the resonances in the hyperbolic layer. The implemented approach using coating with an engineered effective refractive index may find applications in plasmon-enhanced spectroscopy, nanolasers, nonlinearity design, photothermal conversions, and hot-electron generation.