Dark matter (DM) scattering with nuclei in solid-state systems may produce elastic nuclear recoil at high energies and single-phonon excitation at low energies. When the DM momentum is comparable to the momentum spread of nuclei bound in a lattice, q_{0}=sqrt[2m_{N}ω_{0}] where m_{N} is the mass of the nucleus and ω_{0} is the optical phonon energy, an intermediate scattering regime characterized by multiphonon excitations emerges. We study a greatly simplified model of a single nucleus in a harmonic potential and show that, while the mean energy deposited for a given momentum transfer q is equal to the elastic value q^{2}/(2m_{N}), the phonon occupation number follows a Poisson distribution and thus the energy spread is ΔE=qsqrt[ω_{0}/(2m_{N})]. This observation suggests that low-threshold calorimetric detectors may have significantly increased sensitivity to sub-GeV DM compared to the expectation from elastic scattering, even when the energy threshold is above the single-phonon energy, by exploiting the tail of the Poisson distribution for phonons above the elastic energy. We use a simple model of electronic excitations to argue that this multiphonon signal will also accompany ionization signals induced from DM-electron scattering or the Migdal effect. In well-motivated models where DM couples to a heavy, kinetically mixed dark photon, we show that these signals can probe experimental milestones for cosmological DM production via thermal freeze-out, including the thermal target for Majorana fermion DM.