At the highest levels of pulsar timing precision achieved to date, experiments are limited by noise intrinsic to the pulsar. This stochastic wideband impulse modulated self-noise (SWIMS) limits pulsar timing precision by randomly biasing the measured times of arrival and thus increasing the root mean square (rms) timing residual. We discuss an improved methodology of removing this bias in the measured times of arrival by including information about polarized radiation. Observations of J0437-4715 made over a one-week interval at the Parkes Observatory are used to demonstrate a nearly 40 per cent improvement in the rms timing residual with this extended analysis. In this way, based on the observations over a 64 MHz bandwidth centred at 1341 MHz with integrations over 16.78 s we achieve a 476 ns rms timing residual. In the absence of systematic error, these results lead to a predicted rms timing residual of 30 ns in one hour integrations; however the data are currently limited by variable Faraday rotation in the Earth's ionosphere. The improvement demonstrated in this work provides an opportunity to increase the sensitivity in various pulsar timing experiments, for example pulsar timing arrays that pursue the detection of the stochastic background of gravitational waves. The fractional improvement is highly dependent on the properties of the pulse profile and the stochastic wideband impulse modulated self-noise of the pulsar in question. ; Comment: 11 pages, 8 figures, accepted for publication in MNRAS