Private Simultaneous Messages (PSM) is a kind of secure multiparty computation with minimal interaction pattern and minimal security requirement. A PSM protocol is said to be with universal reconstruction for a given function family if the algorithm of the referee (the output party) is independent of a function to be computed and the referee cannot infer the function from a protocol execution. In a recent work by Eriguchi and Shinagawa (EUROCRYPT 2025), the authors proposed a compiler to obtain a PSM protocol for symmetric functions from PSM protocols with universal reconstruction for symmetric functions with smaller domains. The authors also constructed the latter PSM protocols with universal reconstruction, by which the former PSM protocol achieves communication complexity better than the previously known protocols. In this paper, we construct the latter PSM protocols with universal reconstruction for symmetric functions more efficiently; the communication complexity is exponentially (in the input range) smaller than the protocols by Eriguchi and Shinagawa. As a consequence, we also obtain a PSM protocol (and also an ad-hoc PSM protocol and a robust PSM protocol) for symmetric functions that is more efficient than their protocol. Technically, a main ingredient of their protocols is a linear and injective encoding of histograms for the input elements, and our improvement is realized by finding a more efficient encoding of the histograms.