  We broaden the study of the statistical physics of the spin-$S$ Blume-Capel
model with ferromagnetic mean-field interactions $J$ in competition with
short-range antiferromagnetic interactions $K$ in a linear chain in the
thermodynamic limit. This work describes the critical behavior of the model
when the $S$ takes a half integer and an integer value. In both cases the phase
diagrams exhibit new ferromagnetic phases (for certain values of $K$) enclosed
by branches emerging from the first-order frontiers of the pure ferromagnetic
model. For finite temperatures the complex topologies were obtained by
numerical minimization of the free energy.
