In a recent article [Pietropaolo et al., Phys. Rev. Lett.100, 127802 (2008)] we discussed the results of a deep inelastic neutron scattering experiment aimed at the measure of mean kinetic energy, <E(k)>, and momentum distribution of protons in supercooled water. The observed excess of proton mean kinetic energy, with respect to theoretical predictions and measurements in water stable liquid and solid phases, suggested a possible link between the anomalous temperature dependence of water density and the temperature dependence of <E(k)>. Nevertheless, the limited number of data did not allow a more quantitative description. Here we report on new measurements, above and below the temperature of maximum density and in the supercooled phase. These show two <E(k)> maxima: The first one, in the supercooled phase in the range of 269-272 K, and a second one at 277 K, the temperature of the maximum density of water.