Recently, the upper bounds on the static time-reversal (𝒯) and spatial parity (𝒫)-violating electron electric dipole moment (eEDM) and dimensionless constant, characterizing the strength of the 𝒯,𝒫-violating scalar–pseudoscalar nucleus–electron interaction, have been updated in the JILA experiment using the HfF+ cations. We considered two other sources of the 𝒯,𝒫 -violation in HfF+–axion-like-particle (ALP)-mediated scalar–pseudoscalar electron–electron and nucleus–electron interactions. To estimate the magnitude of effects, induced by such interactions in HfF+ we have developed and applied a method which implies the direct use of the ab initio relativistic coupled cluster theory to calculate molecular parameters that characterize the interactions. Using these parameters, we showed that an order of magnitude updated laboratory constraints on the ALP-mediated electron–electron and nucleus–electron interactions can be derived from the experimental data on 𝒯,𝒫-violating effects in HfF+ for a wide range of ALP masses.