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We propose a strategy to potentially best enhance interfacial thermal transport through solid-solid interfaces by adding nano-engineered, exponentially mass-graded intermediate layers. This exponential design rule results in a greater enhancement than a linearly mass-graded interface. By combining calculations using non-equilibrium Green's functions (NEGF) and non-equilibrium molecular dynamics (NEMD), we investigated the role of impedance matching and anharmonicity in the enhancement in addition to geometric parameters such as the number of layers and the junction thickness. Our analysis shows that the effect on thermal conductance is dominated by the phonon thermalization through anharmonic effects, while elastic phonon transmission and impedance matching play a secondary role. In the harmonic limit, increasing the number of layers results in greater elastic phonon transmission at each individual boundary, countered by the decrease of available conducting channels. Consequently, conductance initially increases with number of layers due to improved bridging, but quickly saturates. The presence of slight anharmonic effects (at very low temperature, T = 2 K) turns the saturation into a monotonically increasing trend. Anharmonic effects can further facilitate interfacial thermal transport through the thermalization of phonons at moderate temperatures. At high temperature, however, the role of anharmonicity as a facilitator of interfacial thermal transport reverses. Strong anharmonicity introduces significant intrinsic resistance, overruling the enhancement in thermal conduction at the boundaries. It follows that at a particular temperature, there exists a corresponding junction thickness at which thermal conductance is maximized.