We examine the heat transfer between a sintered metallic powder and liquid 3He at ultralow temperatures. With the combination of a renormalization-group approach with the Lanczos method, a microscopic model is analyzed to yield the low-frequency density of states ρ(ω) of the sinter. In the weak-coupling limit ρ(ω) possesses a pronounced low-frequency peak leading to an enhanced heat transfer coefficient at low temperatures. This enhancement is shown to occur when the ratio of the long-wavelength velocity of sound Cs in the sinter to its velocity Cb in the bulk metal is small and can be optimized by minimizing this ratio.