Aims. We investigate electron temperature (T-e) and gas-phase oxygen abundance (Z(Te)) measurements for galaxies in the local Universe (z& x2004;< & x2004;0.25). Our sample comprises spectra from a total of 264 emission-line systems, ranging from individual HII regions to whole galaxies, including 23 composite HII regions from star-forming main sequence galaxies in the MaNGA survey. Methods. We utilise 130 of these systems with directly measurable T-e(OII) to calibrate a new metallicity-dependent T-e(OIII)-T-e(OII) relation that provides a better representation of our varied dataset than existing relations from the literature. We also provide an alternative T-e(OIII)-T-e(NII) calibration. This new T-e method is then used to obtain accurate Z(Te) estimates and form the mass - metallicity relation (MZR) for a sample of 118 local galaxies. Results. We find that all the T-e(OIII)-T-e(OII) relations considered here systematically under-estimate Z(Te) for low-ionisation systems by up to 0.6 dex. We determine that this is due to such systems having an intrinsically higher O+ abundance than O++ abundance, rendering Z(Te) estimates based only on [OIII] lines inaccurate. We therefore provide an empirical correction based on strong emission lines to account for this bias when using our new T-e(OIII)-T-e(OIII) and T-e(OIII)-T-e(NII) relations. This allows for accurate metallicities (1 sigma& x2004;=& x2004;0.08 dex) to be derived for any low-redshift system with an [OIII]lambda 4363 detection, regardless of its physical size or ionisation state. The MZR formed from our dataset is in very good agreement with those formed from direct measurements of metal recombination lines and blue supergiant absorption lines, in contrast to most other T-e-based and strong-line-based MZRs. Our new T-e method therefore provides an accurate and precise way of obtaining Z(Te) for a large and diverse range of star-forming systems in the local Universe.