Based on the spectral divide-and-conquer algorithm by Nakatsukasa and Higham [SIAM J. Sci. Comput., 35(3):A1325-A1349, 2013], we propose a new algorithm for computing all the eigenvalues and eigenvectors of a symmetric banded matrix with small bandwidth, with the eigenvectors given implicitly as a product of orthonormal matrices stored in the so-called hierarchically off-diagonal low-rank (HODLR) format. For this purpose, we combine our previous work on the fast computation of spectral projectors in the HODLR format, with a novel technique for extracting a basis for the range of such a HODLR matrix. Preliminary numerical experiments demonstrate that our algorithm exhibits quasi-linear complexity for matrices that can be efficiently represented in the HODLR format throughout the divide-and-conquer algorithm, and allows for conveniently dealing with such large-scale matrices.