We calculate the single-particle spectral function for the Hubbard model within the framework of a projection technique equivalent to the two-pole approximation. We show that the two-pole approximation can be well understood as an average characterization of the upper and the lower Hubbard bands, rather than describing a dispersive quasiparticle. By comparing with numerical spectra of finite Hubbard rings and of a 4×4 cluster [P. W. Leung et al., Phys. Rev. B 46, 11 779 (1992)], we show that the present approximation is capable of reproducing essential properties of the single-particle spectral function. In particular, the two-pole spectrum is characterized by a direct gap, in agreement with the exact spectrum. We emphasize the role of local antiferromagnetic correlations.