Hybrid organizations are often confronted with the risk of mission drift, when the social purpose is somehow lost by the evolution of activities and the practices become decouple from the formal purpose. But if the purpose is innovative, tensions appear between the need to develop learning processes through innovations and the risks of losing the social purpose. The article aims at characterizing what are the potential shifts in an organization with an innovative purpose, and whether they constitute a mission drift or not. It is based on a longitudinal case study with a French hybrid organization with an innovative purpose in vocational training. Based on this empirical study, we first confirm that the innovative purpose leads to shifts in activities that are not necessarily mission drifts. We then show that the shifts can be classified in two main categories: those where learning creates a dissension on the interpretation of the organization's purpose, which we call "coverage shift", and those where exploration threatens the commonly accepted interpretation of a purpose, which we call "reinvention shift". These two different situations create differentiated risks regarding mission drift, and therefore call for differentiated response at the governance level. This study thus contributes to enriching the literature on the mission drift in hybrid organizations. It also suggests some avenues how to deviation control can be complemented by a capacity to assess the innovative shifts consistent with the purpose.