Analyses of Type Ia supernovae (SNe Ia) have found puzzling correlations between their standardized luminosities and host galaxy properties: SNe Ia in high-mass, passive hosts appear brighter than those in lower mass, star-forming hosts. We examine the host galaxies of SNe Ia in the Dark Energy Survey 3-yr spectroscopically confirmed cosmological sample, obtaining photometry in a series of ‘local’ apertures centred on the SN, and for the global host galaxy. Here, we study the differences in these host galaxy properties, such as stellar mass and rest-frame U – R colours, and their correlations with SN Ia parameters including Hubble residuals. We find all Hubble residual steps to be >3σ in significance, both for splitting at the traditional environmental property sample median and for the step of maximum significance. For stellar mass, we find a maximal local step of 0.098 ± 0.018 mag; ~0.03 mag greater than the largest global stellar mass step in our sample (0.070 ± 0.017 mag). When splitting at the sample median, differences between local and global U – R steps are small, both ~0.08 mag, but are more significant than the global stellar mass step (0.057 ± 0.017 mag). We split the data into sub-samples based on SN Ia light-curve parameters: stretch (x1) and colour (c), finding that redder objects (c > 0) have larger Hubble residual steps, for both stellar mass and U – R, for both local and global measurements, of ~0.14 mag. Additionally, the bluer (star-forming) local environments host a more homogeneous SN Ia sample, with local U – R rms scatter as low as 0.084 ± 0.017 mag for blue (c < 0) SNe Ia in locally blue U – R environments.