The author reformulates the weighted-density approximation based on a global average of the density by using the unified density-functional approach for non-uniform classical fluids which was introduced by Zeng and Oxtoby (1990). He shows that the modified weighted-density approximation (MWDA) of Denton and Ashcroft (1991) and the weighted-density approximation of Zeng and Oxtoby can be approximated as second-order truncations of exact functional expansions. Through this reformulation, a new version of the modified weighted-density approximation is proposed (NMWDA) depending on the charging parameter is proposed and a basic question arising from the density-functional expansions is briefly discussed. It is shown that for homogeneous systems the NMWDA and the weighted-density approximation including the higher-order contributions also lead to the usual hypernetted-chain equation of state, and the homogeneous properties of the weighted-density approximation do not depend on the detailed forms of n-particle hierarchy functions. Finally, the author applies the NMWDA to the liquid-solid transition. The numerical results obtained are compared with those of computer simulations and other approximations and show good agreement with computer simulations.