  We investigate electron temperature (Te) and gas-phase oxygen abundance
(Z_Te) measurements for galaxies in the local Universe (z < 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. We utilise 130
of these systems with directly measurable Te(OII) to calibrate a new
metallicity-dependent Te(OIII) - Te(OII) relation that provides a better
representation of our varied dataset than existing relations from the
literature. We also provide an alternative Te(OIII) - Te(NII) calibration. This
new Te method is then used to obtain accurate Z_Te estimates and form the mass
- metallicity relation (MZR) for a sample of 118 local galaxies. We find that
all the Te(OIII) - Te(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 Te(OIII) - Te(OII)
and Te(OIII) - Te(NII) relations. This allows for accurate metallicities
(1sigma = 0.08 dex) to be derived for any low-redshift system with an
[OIII]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 Te-based and strong-line-based MZRs. Our new
Te 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.
