In this paper, we propose a lossy source coding approach to improve embedding efficiency in steganography. A higher embedding efficiency (decreasing the distortion function) is desirable since it leads to better security. We propose to use a soft-hard belief propagation guided decimation (BPGD) algorithm for the encoding problem with low-density generator matrix (LDGM) codes. However, for good distortion performance, the parameters of the soft or soft-hard BPGD need to be tuned. To achieve this, we apply the cavity method to predict a value called the dynamical phase transition, which can minimize the distortion function for the soft-hard BPGD. This approach facilitates secure steganography by finding optimal parameters for the distortion function without the need for exhaustive search and simulation. Our method is shown to outperform related works in terms of embedding efficiency, performance, and complexity.