In a recent paper (Barros, Sousa in: Kodai Math. J. 2009) the authors proved that closed oriented non-totally geodesic minimal hypersurfaces of the Euclidean unit sphere have index of stability greater than or equal to n + 3 with equality occurring at only Clifford tori provided their second fundamental forms A satisfy the pinching: |A|2 ≥ n. The natural generalization for this pinching is −(r + 2)S r+2 ≥ (n − r)S r > 0. Under this condition we shall extend such result for closed oriented hypersurface Σ n of the Euclidean unit sphere $${\mathbb{S}^{n+1}}$$ with null S r+1 mean curvature by showing that the index of r-stability, $${Ind_{\Sigma^n}^{r}}$$ , also satisfies $${Ind_{\Sigma^n}^{r}\ge n+3}$$ . Instead of the previous hypothesis if we consider $${\frac{S_{r+2}}{{S_r}}}$$ constant we have the same conclusion. Moreover, we shall prove that, up to Clifford tori, closed oriented hypersurfaces $${\Sigma^{n}\subset \mathbb{S}^{n+1}}$$ with S r+1 = 0 and S r+2 < 0 have index of r-stability greater than or equal to 2n + 5.