© 2018 American Physical Society. Several approaches to linear-scaling density functional theory (LS-DFT) that seek to achieve accuracy equivalent to plane-wave methods do so by optimizing in situ a set of local orbitals in terms of which the density matrix can be accurately expressed. These local orbitals, which can also accurately represent the canonical Kohn-Sham orbitals, qualitatively resemble the maximally localized Wannier functions employed in band structure interpolation. As LS-DFT methods are increasingly being used in real-world applications demanding accurate band structures, it is natural to question the extent to which these optimized local orbitals can provide sufficient accuracy. In this paper, we present and compare, in principle and in practice, two methods for obtaining band structures. We apply these to a (10, 0) carbon nanotube as an example. By comparing with the results from a traditional plane-wave pseudopotential calculation, the optimized local orbitals are found to provide an excellent description of the occupied bands and some low-lying unoccupied bands, with consistent agreement across the Brillouin zone. However free-electron-like states derived from weakly bound states independent of the σ and π orbitals can only be found if additional local orbitals are included.