  We revisit the lattice formulation of the Schwinger model using the
Kogut-Susskind Hamiltonian approach with staggered fermions. This model,
introduced by Banks et al., contains the mass term $m_{\rm lat} \sum_{n}
(-1)^{n} \chi^\dagger_n \chi_n$, and setting it to zero is often assumed to
provide the lattice regularization of the massless Schwinger model. We instead
argue that the relation between the lattice and continuum mass parameters
should be taken as $m_{\rm lat}=m- \frac 18 e^2 a$. The model with $m=0$ is
shown to possess a discrete chiral symmetry that is generated by the unit
lattice translation accompanied by the shift of the $\theta$-angle by $\pi$.
While the mass shift vanishes as the lattice spacing $a$ approaches zero, we
find that including this shift greatly improves the rate of convergence to the
continuum limit. We demonstrate the faster convergence using both numerical
diagonalizations of finite lattice systems, as well as extrapolations of the
lattice strong coupling expansions.
