We study high-order harmonic generation in aligned molecules close to the ionization threshold. Two distinct contributions to the harmonic signal are observed, which show very different responses to molecular alignment and ellipticity of the driving field. We perform a classical electron trajectory analysis, taking into account the significant influence of the Coulomb potential on the strong-field-driven electron dynamics. The two contributions are related to primary ionization and excitation processes, offering a deeper understanding of the origin of high harmonics near the ionization threshold. This Letter shows that high-harmonic spectroscopy can be extended to the near-threshold spectral range, which is in general spectroscopically rich.