The theory of the equation of state formulated earlier and based on excluded volume analysis was extended to the entire density range by using higher-order approximations. The equations were derived to satisfy the requirement of maximum simplicity and contained a single adjustment parameter found by numerically simulating a fluid of hard spheres. The accuracy of the equation of state was raised to 0.7% over the entire density range. Simultaneously, the best-known modern equations of state for hard spheres were reproduced and the place that each of them occupied in the hierarchy of approximations was established.