Astrophysical neutral particles, such as neutrons, can point directly to their sources since they are not affected by magnetic fields. We expect neutron production in the immediate vicinity of the acceleration sites due to cosmic ray interactions. Hence, a high-energy neutron flux could help to identify sources of cosmic rays in the EeV range. Free neutrons, although unstable, can travel a mean distance of 9.2 kpc times their energy in EeV. Due to the neutron instability, we limit the searches to Galactic candidate sources. Since air showers initiated by a neutron are indistinguishable from those generated by a proton, we would recognize a neutron flux as an excess of events from the direction of its source. Previous searches using events with a zenith angle up to $60^{\circ}$ and energies above 1 EeV found no surplus of events that would indicate a neutron flux. We present the results of the search for evidence of high-energy neutron fluxes using a data set about three times larger than the previous work. We investigate the sky in the field of view of the Pierre Auger Observatory, narrowing down to specific directions of candidate sources. With respect to previous works, we extend the angular range up to zenith angles of $80^{\circ}$, reaching declinations from $-90^{\circ}$ to $+45^{\circ}$, and the energy range going as low as 0.1 EeV. The extension in the field of view provides exposure to the Crab Nebula for the first time.