We propose self-consistent solutions to the Ornstein-Zemike equation where the approximate closure is replaced by a bridge function expansion, whose main advantage is the improvement of correlation functions. Unknown coefficients of this expansion are found from the principle of total thermodynamic consistency. The latter includes not only the conventional pressure-compressibility equation but also the relation between internal energy and pressure. We show that utilizing only the first equation one may face a nonunique partially consistent solution conditioned by noncomplete formulation of the consistency problem. At the same time the suggested set of equations is sufficient to determine a true and unique physical solution regardless of the number of unknown coefficients. In this paper we expand the bridge function in powers of potential of mean force and perform the example of building the approximate self-consistent closure. Moreover, the approach via total thermodynamic consistency introduces the value of residual inconsistency as the internal criterion of accuracy, which is equal to zero only for the exact closure relation and the corresponding solution. This value is found to be in agreement with the deviation of thermodynamic quantities from numeric simulation data. The proposed method is tested on the classical model of a Lennard-Jones fluid in a wide range of temperatures and high densities.