Context. Accurate polarimetric calibration of the radio pulse profiles from pulsars is crucial for studying their radiation properties at these wavelengths. Additionally, inaccurate calibration can distort recorded pulse profiles, introducing noise in time of arrival (TOA) data and thus degrading pulsar timing analyses. One method for determining the full polarimetric response of a given telescope is to conduct observations of bright polarized pulsars over wide ranges of parallactic angles, to sample different orientations of their polarization angle and in turn determine the cross-couplings between polarization feeds. Aims. The Nançay decimetric Radio Telescope (NRT) is a 94 m equivalent meridian telescope, capable of tracking a given pulsar for approximately 1 h around transit. The NRT therefore cannot sample wide ranges of parallactic angles when observing a given pulsar, so until late 2019 the polarimetric calibration of 1.4 GHz pulsar observations with the NRT was rudimentary. We therefore aimed to develop a method for improving the calibration of NRT observations, overcoming the above-mentioned limitation. Ultimately, our goal was to improve the quality of NRT pulsar timing, with better calibrated pulsar pulse profiles. Methods. In November 2019, we began conducting regular observations of the bright and highly linearly polarized pulsar PSR J0742−2822, in a special observing mode in which the feed horn rotates by ~180° over the course of the 1 h observation, mimicking wide parallactic angle variations and in principle enabling us to determine the polarimetric response of the NRT at 1.4 GHz. In addition, we assessed the quality of the NRT timing of a selection of millisecond pulsars (MSPs), namely, J1730−2304, J1744−1134, and J1857+0953, with conventional TOAs extracted from total intensity pulse profiles, and TOAs extracted with the Matrix Template Matching (MTM) technique, designed to compensate for putative polarimetric calibration errors. Results. From the analysis of the rotating horn observations of PSR J0742−2822 we could determine the cross-couplings between the polarization feeds and also constrain the Stokes parameters of the noise diode signal, which prior to this work was erroneously assumed to be ideal and was used as the only reference source for the calibration of pulsar observations. The improved polarimetric response of the NRT as determined from these observations was applied to observations of a selection of MSPs with published polarimetric properties. We find that the new polarimetric profiles and polarization position angles are consistent with previous findings, unlike NRT polarimetric results obtained with the previously used method of calibration. The analysis of the timing data shows that the new calibration method improves the quality of the timing, and the MTM method proves very effective at reducing noise from imperfect calibration. For pulsars with sufficient degrees of polarization, the MTM method appears to be the preferred method of extracting TOAs from NRT observations.