We show that a certain amount of non-local overshoot at the base of the outer convective envelope of low mass stars, while climbing along the red giant branch (RGB) toward central He ignition, explains the shift of about 0.4 V mag required to bring the luminosity of the bump expected in the theoretical RGB luminosity function into agreement with the observational data. The extension of the envelope overshoot is about 0.7 Hp, where Hp is the local pressure scale height. In addition to this, we find that an equal amount of overshoot from the convective envelope in intermediate-mass stars greatly affects their subsequent evolution in the C-M diagram, producing extended loops even for models computed with significant overshoot from the convective core. This finding improves upon a weak point of models with core overshoot. In fact, while these models predict correctly the ratio of giant to main sequence stars observed in the C-M diagrams of well studied clusters in the Large Magellanic Cloud (LMC), they hardly fit the extension of the observed blue loops. It is worth recalling that classical models, in which core overshoot is neglected, although reproducing the morphology of the C-M diagrams (loop extension), fail in predicting the correct number of stars (ratio of lifetimes) in various phases across the C-M diagram. These results cast light on the role played by core and envelope overshoot. The former provides the correct H- and He-burning lifetimes for intermediate-mass stars, whereas the latter reproduces the observed morphology of young LMC clusters and the correct location of the bump in the RGB luminosity function of globular clusters. We propose that stellar models ought to be calculated taking into account both envelope overshoot, calibrated in the domain of globular clusters, and core overshoot, in turn calibrated in the range of young LMC clusters. New evolutionary sequences, to be presented elsewhere, incorporating both core and envelope overshoot, are found to agree with the observations much better than classical models (no overshoot of any kind) or models with core overshoot alone.
