Near-field radiative heat transfer exhibits various effects, such as amplification due to the geometry of the system. In this work, we construct a periodic layered structure which consists of multiple layers of alternating materials. Radiative heat transfer (RHT) between nanoparticles placed on each side of an intermediate structure is studied. Thermal energy exchange between nanoparticles is assisted by transmitted evanescent fields, which is theoretically included in the system's Green's function. We show that the resulting heat transfer with the assistance of a multilayered structure is more than five orders of magnitude higher than that in the absence of the multilayered structure at the same interparticle distance. This increase is observed over a broad range of distances ranging from near to far field. This is due to the fact that the intermediate multilayered structure supports hyperbolic phonon polaritons, where the edge frequencies of the type-I and type-II reststrahlen bands coincide at a value that is approximately equal to the nanoparticle resonance. This allows high-k evanescent modes to resonate with the nanoparticles. The effects of the number of layers and fill factor in the multilayered structure on RHT are examined. Finally, we show that when there is a lateral distance between the two particles assisted by the interference of surface waves, RHT conductance exhibits an oscillating and nonmonotonic behavior with respect to the lateral distance between nanoparticles. These results illustrate a powerful method for regulating energy transport in particle systems and can be relevant for effective energy management at nano-micro scales.