The authors use the analytical equation of state obtained by the discrete perturbation theory [A. L. Benavides and A. Gil-Villegas, Mol. Phys. 97, 1225 (1999)] to study the phase diagram of fluids with discrete spherical potentials formed by a repulsive square-shoulder plus an attractive square-well interaction (SS+SW). This interaction is characterized by the usual energy and size parameters plus three dimensionless parameters: two of them measuring the widths of the SS and the SW and the third the relative height of the SS. The matter of interest is that, for certain values of the interaction parameters, the SS+SW systems exhibit more than one first-order fluid-fluid transition. The evidence that several real substances (such as water, phosphorus, carbon, and silica, among others) exhibit an extra liquid-liquid transition has drawn interest into the study of interactions responsible for this behavior. The simple SS+SW fluid is one of the systems that, in spite of being spherically symmetric, shows multiple fluid-fluid transitions. In this work the authors investigate systematically the effect on the phase diagram of varying the interaction parameters. The use of an analytical free-energy equation gives a clear thermodynamic picture of the emergence of different types of critical points, throwing new light on the phase behavior of these fluids and thus clarifying previous results obtained by other techniques. The interplay of attractive and repulsive forces with several scale lengths produces very rich phase diagrams, including cases with three critical points. The region of the interaction-parameter space where multiple critical points appear is mapped for various families of interactions.