In this third paper in a series we compare spectral feature measurements to photometric properties of 108 low-redshift (z < 0.1) Type Ia supernovae (SNe Ia) with optical spectra within 5 d of maximum brightness. We find the pseudo-equivalent width (pEW) of the Si II 4000 line to be a good indicator of light-curve width, and the pEWs of the Mg II and Fe II complexes are relatively good proxies for SN colour. We also employ a combination of light-curve parameters (specifically the SALT2 stretch and colour parameters x_1 and c, respectively) and spectral measurements to calculate distance moduli. The residuals from these models are then compared to the standard model which uses only light-curve stretch and colour. Our investigations show that a distance model that uses x_1, c, and the velocity of the Si II 6355 feature does not lead to a decrease in the Hubble residuals. We also find that distance models with flux ratios alone or in conjunction with light-curve information rarely perform better than the standard (x_1,c) model. However, when adopting a distance model which combines the ratio of fluxes near ~3750 Ang. and ~4550 Ang. with both x_1 and c, the Hubble residuals are decreased by ~10 per cent, which is found to be significant at about the 2-sigma level. The weighted root-mean-square of the residuals using this model is 0.130 +/- 0.017 mag (as compared with 0.144 +/- 0.019 mag when using the same sample with the standard model). This Hubble diagram fit has one of the smallest scatters ever published and at the highest significance ever seen in such a study. Finally, these results are discussed with regard to how they can improve the cosmological accuracy of future, large-scale SN Ia surveys. [Abridged]
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