By using a so-called differential-integral method on the chemical potential of a hard-sphere fluid, a special variant of our previously developed expression that describes the interaction between charged plates immersed in an electrolyte, is introduced to examine the hard-sphere fluid mediated pressure in a slit. The resulting expression consists of a kinetic contribution and a hard-sphere contribution, and it is formulated as a function of the single-particle direct correlation function and the density distribution of a hard-sphere fluid. It allows us to conveniently apply the classic density functional theory to explicitly investigate the influence of the hard-sphere excluded-volume effect on the interaction pressure between surfaces. In this study, a newly proposed weighted correlation approach (WCA)-Denton and Ashcroft (DA) method is employed to predict the interaction pressure as well as its pressure components for a hard-sphere fluid inside a slit pore. Comparisons with the results from the Monte Carlo simulations and the fundamental measure theory suggest that the WCA-DA method is able to accurately capture the detailed characteristic pattern of the pressure-separation curves at different fluid densities. It is also found, both qualitatively and quantitatively, that the hard-sphere pressure contribution dominates over the kinetic pressure contribution in determining the oscillatory behaviour of the interaction pressure curves, especially when a hard-sphere fluid of high density is concerned.