The emergence of multilayer metamaterials in the research field of enhancing spontaneous emission rates has recently received extensive attention. Previous research efforts mostly focus on periodic metal-dielectric multilayers in hyperbolic dispersion region; however, the influence of lattice order in subwavelength multilayers on spontaneous emission is rarely studied. Here, we observe the stronger Purcell enhancement of quantum dots coupled to the aperiodic metal-dielectric multilayer with Thue-Morse lattice order from elliptical to hyperbolic dispersion regions, compared to the periodic multilayer with the same metal filling ratio. This work demonstrates the potential of utilizing quasiperiodic metamaterial nanostructures to engineer the local density of states for various nanophotonic applications.