Let $X$ be a complex smooth Fano variety whose minimal anticanonical degree of non-free rational curves on $X$ is at least $\dim X-2$. We give a classification of extremal contractions of such varieties. As applications, we obtain a classification of Fano fourfolds whose pseudoindex and Picard number are greater than one and study the structure of Fano varieties with nef third exterior power of the tangent bundle.