Obtaining accurate redshifts from mid-infrared (MIR) low-resolution (R ∼ 100) spectroscopy is challenging because the wavelength resolution is too low to detect narrow lines in most cases. Yet, the number of degrees of freedom and diversity of spectral features are too high for regular spectral energy distribution (SED) fitting techniques to be convenient. Here we present a new SED-fitting-based routine for redshift determination that is optimized for MIR low-resolution spectroscopy. Its flexible template scaling increases the sensitivity to slope changes and small-scale features in the spectrum, while a new selection algorithm called maximum combined pseudo-likelihood (MCPL) provides increased accuracy and a lower number of outliers compared to the standard maximum-likelihood (ML) approach. Unlike ML approach, the MCPL approach searches for local (instead of absolute) maxima of a ‘pseudo-likelihood’ (PL) function, and combines results obtained for all the templates in the library to weed out spurious redshift solutions. The capabilities of the MCPL approach are demonstrated by comparing its redshift estimates to those of the regular ML approach and to the optical spectroscopic redshifts of a sample of 491 Spitzer/Infrared Spectrograph spectra from extragalactic sources at 0 < z < 3.7. The MCPL approach achieves a redshift accuracy Δ(z)/(1 + z) < 0.005 for 78 per cent of the galaxies in the sample compared to 68 per cent for the ML approach. The rate of outliers [Δ(z)/(1 + z) > 0.02] is 14 per cent for the MCPL approach and 22 per cent for the ML approach. χ2 values for ML solutions are found to correlate with the signal-to-noise ratio of the spectra, but not with redshift accuracy. By contrast, the peak value of the normalized combined PL (γ) is found to provide a good indication on the reliability of the MCPL solution for individual sources. The accuracy and reliability of the redshifts depend strongly on the MIR SED. Sources with significant polycyclic aromatic hydrocarbon emission obtain much better results compared to sources dominated by active galactic nucleus continua. Nevertheless, for a given γ the frequency of accurate solutions and outliers is largely independent of their SED type. This reliability indicator for MCPL solutions allows to select subsamples with highly reliable redshifts. In particular, a γ > 0.15 threshold retains 79 per cent of the sources with Δ(z)/(1 + z) < 0.005 while reducing the outlier rate to 3.8 per cent.