We consider models of inflation in supergravity with a shift symmetry. We focus on models with one moduli and one inflaton field. The presence of this symmetry guarantees the existence of a flat direction for the inflaton field. Mildly breaking the shift symmetry using a superpotential which depends not only on the moduli but also on the inflaton field allows one to lift the inflaton flat direction. Along the inflaton direction, the η-problem is alleviated. Combining the KKLT mechanism for moduli stabilization and a shift symmetry breaking superpotential of the chaotic inflation type, we find models reminiscent of "mutated hybrid inflation" where the inflationary trajectory is curved in the moduli-inflaton plane. We analyze the phenomenology of these models and stress their differences with both chaotic and hybrid inflation.