Complex-valued conjugate symmetric Hadamard transform (C-CSHT) is a variant of complex Hadamard transform and effective for some signal processing and communication applications. Its closed-form factorization of the general N-channel (N = 2 m) case was recently proposed, however, there still exist a room to find an effective factorization especially for unified factorization of C-CSHT and its real-valued transform counterpart (R-CSHT). In this paper, we present another simple closed-form factorization of C-CSHT based on that of R-CSHT. The proposed factorization is applicable for both complex- and real-valued CSHTs with one factorization. Furthermore, the relationship with the common block transform, DCT, is revealed.