  Context: 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. Aims: We address this question by
improving the interpolator of the MILES (Medium-resolution INT Library of
Empirical Spectra) library in the low effective temperature regime (Tefff <
4800 K), and we refine the determination of the parameters of the cool MILES
stars. Methods: We use the ULySS package to determine the atmospheric
parameters (Teff, logg and [Fe/H]), and measure the biases of the results with
respect to our updated compilation of parameters calibrated against theoretical
spectra. After correcting some systematic effects, we compute a new
interpolator that we finally use to redetermine the atmospheric parameters
homogeneously and assess the biases. Results: 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 = 2900 K compared to 3500 K previously. The
mean residual biases on Teff, logg, and [Fe/H], with respect to the literature
compilation for the coolest stars (Teff <= 3800 K) computed using the new
interpolator, are -15 K, -0.02 dex, and 0.02 dex respectively. The
corresponding estimations of the external precision are 63 K, 0.23 dex, and
0.15 dex respectively. For the stars with Teff in the range 3800 - 4200 K, the
determinations of Teff and [Fe/H] have been slightly improved. At higher
temperatures, the new interpolator is comparable to the original one. The new
version of the interpolator is publicly available.
