Astrophysical neutral particles, such as neutrons, are produced through interactions of charged cosmic rays in the vicinity of their acceleration sites and are not deflected by magnetic fields during propagation. Therefore, they can be traced back to their sources. Despite being unstable particles, neutrons can travel approximately 9.2 kpc per EeV of energy before decaying, confining the search for their possible sources to the Milky Way. In this study, which was carried out with 19 years of data collected by the Pierre Auger Observatory, we analysed data sets nearly three times larger than those used in previous analyses. We extended the search to declinations up to $+45^\circ$ and to lower energy showers by including data sets with reconstructed primary energies down to 0.1 EeV. This extensive, high-quality dataset is studied in correlation with catalogues of over 800 Galactic candidate sources, including the Crab Nebula, which is studied for the first time in this type of analysis. The analysis method we present has been specifically developed for this study, allowing us to establish upper limits on the neutron flux from the sources under investigation.