We present results from high-resolution cosmological hydrodynamical simulations of a Milky-Way-sized halo, aimed at studying the effect of feedback on the nature of gas accretion. Simulations include a model of interstellar medium and star formation, in which supernova (SN) explosions provide effective thermal feedback. We distinguish between gas accretion onto the halo, which occurs when gas particles cross the halo virial radius, and gas accretion onto the central galaxy, which takes place when gas particles cross the inner one-tenth of the virial radius. Gas particles can be accreted through three different channels, depending on the maximum temperature value, T max, reached during the particles' past evolution: a cold channel for T max < 2.5 × 105 K, a hot one for T > 106 K, and a warm one for intermediate values of T max. We find that the warm channel is at least as important as the cold one for gas accretion onto the central galaxy. This result is at variance with previous findings that the cold mode dominates gas accretion at high redshift. We ascribe this difference to the different SN feedback scheme implemented in our simulations. While results presented so far in the literature are based on uneffective SN thermal feedback schemes and/or the presence of a kinetic feedback, our simulations include only effective thermal feedback. We argue that observational detections of a warm accretion mode in the high-redshift circumgalactic medium would provide useful constraints on the nature of the feedback that regulates star formation in galaxies.