The enhancement of thermal conductivity via surface electromagnetic waves (SEWs) supported in nanostructures has recently attracted attention as a remedy for issues raised due to the reduction of thermal conductivity in nanoscale confinement. Among them, multilayered structures on a substrate are prevalent in nanosized systems, such as electronic nanodevices, meaning that analysis on those structures is indispensable. In this work, three basic multilayered structures are selected and the analytical expressions for SEWs supported in each structure are derived. This analytical approach enables us to figure out which factors are crucial for enhancing SEW thermal conductivity using multilayers. It is also found that the solution can be extended to various materials and provides guidelines on which configurations are desirable for increasing the thermal conductivity. Furthermore, the analytical solutions reduce the calculation time significantly such that the optimal configuration, which can yield a SEW thermal conductivity of 1.27 W/mK, corresponding to 90% of the thermal conductivity of bulk glass, is found using a genetic algorithm. This study thus provides a method for efficiently managing thermal issues in nanosized devices.