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Context.Type Ia supernovae (SNe Ia) are a key probe in modern cosmology, as they can be used to measure luminosity distances at gigaparsec scales. Models of their light curves are used to project heterogeneous observed data onto a common basis for analysis. Aims.The SALT model currently used for SN Ia cosmology describes SNe as having two sources of variability, accounted for by a color parameterc, and a “stretch” parameterx₁. We extend the model to include an additional parameter we labelx₂, to investigate the cosmological impact of currently unaddressed light-curve variability. Methods.We constructed a new SALT model, that we dub “SALT3+”. This model was trained by an improved version of theSALTshakercode, using training data combining a selection of the second data release of cosmological SNe Ia from the Zwicky Transient Facility and the existing SALT3 training compilation. Results.We find additional, coherent variability in supernova light curves beyond SALT3. Most of this variation can be described as phase-dependent variation ing − randr − icolor curves, correlated with a boost in the height of the secondary maximum ini-band. These behaviors correlate with spectral differences, particularly in line velocity. We find that fits with the existing SALT3 model tend to address this excess variation with the color parameter, leading to less informative measurements of supernova color. We find that neglecting the new parameter in light-curve fits leads to a trend in Hubble residuals withx₂of 0.039 ± 0.005 mag, representing a potential systematic uncertainty. However, we find no evidence of a bias in current cosmological measurements. Conclusions.We conclude that extended SN Ia light-curve models promise mild improvement in the accuracy of color measurements, and corresponding cosmological precision. However, models with more parameters are unlikely to substantially affect current cosmological results.