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Bifurcation analysis of confined salt-finger convection using single-mode equations obtained from a severely truncated Fourier expansion in the horizontal is performed. Strongly nonlinear staircase-like solutions having, respectively, one (S1), two (S2) and three (S3) regions of mixed salinity in the vertical direction are computed using numerical continuation, and their stability properties are determined. Near onset, the one-layer S1 solution is stable and corresponds to maximum salinity transport among the three solutions. The S2 and S3 solutions are unstable but exert an influence on the statistics observed in direct numerical simulations (DNS) in larger two-dimensional (2D) domains. Secondary bifurcations of S1 lead either to tilted-finger (TF1) or to traveling wave (TW1) solutions, both accompanied by the spontaneous generation of large-scale shear, a process favored for lower density ratios and Prandtl numbers (Pr). These states at low Pr are associated, respectively, with two-layer and three-layer staircase-like salinity profiles in the mean. States breaking reflection symmetry in the midplane are also computed. In 2D and for low Pr the DNS results favor direction-reversing tilted fingers (RTF) resembling the pulsating wave state observed in other systems. Two-layer and three-layer mean salinity profiles corresponding to RTF and TW1 are observed in 2D DNS averaged over time. The single-mode solutions close to the high wavenumber onset are in an excellent agreement with 2D DNS in small horizontal domains and compare well with 3D DNS.