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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 …