We explore the phase diagram of a lattice fermion model that shows two distinct mechanisms of mass generation from a massless phase. The first is the standard spontaneous symmetry breaking (SSB) paradigm induced by a fermion bilinear condensate; and the other is the mechanism of symmetric mass generation (SMG). Using the fermion-bag Monte Carlo method on large cubical lattices, we find evidence for traditional second-order critical points separating the first two and the latter two phases. Remarkably, these critical points appear to merge at a multicritical point with enhanced symmetry when the symmetry-breaking parameter is tuned to zero, giving rise to the recently discovered direct second-order transition between the massless and symmetric massive fermion phases.