  In this work we analyze the finite-volume and magnetic effects on the phase
structure of a generalized version of Nambu--Jona-Lasinio model with three
quark flavors. By making use of mean-field approximation and Schwinger's
proper-time method in a toroidal topology with antiperiodic conditions, we
investigate the gap equation solutions under the change of the size of
compactified coordinates, strength of magnetic field, temperature and chemical
potential. The 't Hooft interaction contributions are also evaluated. The
thermodynamic behavior is strongly affected by the combined effects of relevant
variables. The findings suggest that the broken phase is disfavored due to both
increasing of temperature and chemical potential, and the drop of the cubic
volume of size $L$, whereas it is stimulated with the augmentation of magnetic
field. In particular, the reduction of $L$ (remarkably at $L\approx 0.5 - 3
$~fm) engenders a reduction of the constituent masses for $u,d,s$-quarks
through a crossover phase transition to the their corresponding current quark
masses. On the other hand, the presence of a magnetic background generates
greater values constituent quark masses, inducing smaller sizes and greater
temperatures at which the constituent quark masses drop to the respective
current ones.
