We study the problem of transmitting classical information using quantum Gaussian states on a family of phase-noise channels with a finite decoherence time, such that the phase-reference is lost after m consecutive uses of the transmission line. This problem is relevant for long-distance communication in free space and optical fiber, where phase noise is typically considered as a limiting factor. The Holevo capacity of these channels is always attained with photon-number encodings, challenging with current technology. Hence for coherent-state encodings the optimal rate depends only on the total-energy distribution and we provide upper and lower bounds for all m, the latter attainable at low energies with on/off modulation and photodetection. We generalize this lower bound to squeezed-coherent encodings, exhibiting for the first time to our knowledge an unconditional advantage with respect to any coherent encoding for m = 1 and a considerable advantage with respect to its direct coherent counterpart for m > 1. This advantage is robust with respect to moderate attenuation, and persists in a regime where Fock encodings with up to two-photon states are also suboptimal. Finally, we show that the use of part of the energy to establish a reference frame is sub-optimal even at large energies. Our results represent a key departure from the case of phase-covariant Gaussian channels and constitute a proof-of-principle of the advantages of using non-classical, squeezed light in a motivated communication setting.