In this work, we study the magnitude–redshift relation of a non-standard cosmological model. The model under consideration was first investigated within a special case of metric-affine gravity (MAG) and was recently recovered via different approaches by two other groups. Apart from the usual cosmological parameters for pressureless matter Ω_m, cosmological constant/dark energy Ω_λ and radiation Ω_r a new density parameter Ω_ψ emerges. The field equations of the model reduce to a system which is effectively given by the usual Friedmann equations of general relativity, supplied by a correction to the energy density and pressure in the form of Ω_ψ, which is related to the non-Riemannian structure of the underlying spacetime. We search for the best-fit parameters using recent SN Ia data sets and constrain the possible contribution of a new dark-energy-like component at low redshifts, thereby putting an upper limit on the presence of non-Riemannian quantities in the late stages of the universe. In addition, the impact of placing the data in redshift bins of variable size is studied. The numerical results of this work also apply to several anisotropic cosmological models which, on the level of the field equations, exhibit a similar scaling behaviour of the density parameters like our non-Riemannian model.