The full spectrum fitting of stellar spectra against a library of empirical spectra is a well-established approach to measure the atmospheric parameters of FGK stars with a high internal consistency. Extending it towards cooler stars still remains a challenge. We address this question by improving the MILES interpolator (Prugniel et al., 2011, Cat. J/A+A/531/A165; hereafter PVK) in the low effective temperature regime (Teff<4800K). We measure the biases of the determined parameters with respect to our updated compilation of parameters. After correcting some systematic effects, we compute a new interpolator that we finally use to redetermine the atmospheric parameters homogeneously and assess the biases. Based on an updated literature compilation, we determine Teff in a more accurate and unbiased manner compared to those determined with the original interpolator. The validity range is extended downwards to about Teff=2900K compared to 3500K previously. The measured parameters are restricted to the current sub sample of MILES, and we have shown that they are generally more reliable than those presented in PVK. But the small discontinuity between the PVK and new measurements is significant, in particular for Teff, and should be corrected if one needs to join the two series.