In this third paper in a series we compare spectral feature measurements to photometric properties of 108 low-redshift (z < 0.1, 〈z〉 ≈ 0.023) Type Ia supernovae (SNe Ia) for which we have optical spectra within 5 d of maximum brightness. The spectral data were obtained from 1989 to the end of 2008 as part of the Berkeley Supernova Ia Program (BSNIP) and are presented in BSNIP I by Silverman et al., and the photometric data come mainly from the Lick Observatory Supernova Search and are published by Ganeshalingam et al. The spectral measurements are presented and discussed in BSNIP II by Silverman, Kong & Filippenko, and the light-curve fits and photometric parameters can be found in Ganeshalingam et al. (in preparation). A variety of previously proposed correlations between spectral and photometric parameters are investigated using the large and self-consistent BSNIP data set. We find the pseudo-equivalent width (pEW) of the Si ii λ4000 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 Spectral Adaptive Light-curve Template 2 stretch and colour parameters x1 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 x1, c and the velocity of the Si ii λ6355 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 (x1, c) model. However, when adopting a distance model which combines the ratio of fluxes near ∼3750 and 4550 Å with both x1 and c, the Hubble residuals are decreased by ∼10 per cent, which is found to be significant at about the 2σ 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.