We report on a search for a new, short-range, spin-dependent interaction using a modified version of the experimental apparatus used to measure the permanent neutron electric dipole moment at the Paul Scherrer Institute. This interaction, which could be mediated by axion-like particles, concerned the unpolarized nucleons (protons and neutrons) near the material surfaces of the apparatus and polarized ultracold neutrons stored in vacuum. The dominant systematic uncertainty resulting from magnetic-field gradients was controlled to an unprecedented level of approximately 4 pT cm^-1 using an array of optically-pumped cesium vapor magnetometers and magnetic-field maps independently recorded using a dedicated measurement device. No signature of a theoretically predicted new interaction was found, and we set a new limit on the product of the scalar and the pseudoscalar couplings g_sg_pλ² < 8.3 x 10^-28 m² (95% C.L.) in a range of 5  µm  < λ< 25  mm for the monopole-dipole interaction. This new result confirms and improves our previous limit by a factor of 2.7 and provides the current tightest limit obtained with free neutrons.