  For $n\geq 4$, let $X$ be a complex smooth Fano $n$-fold whose minimal
anticanonical degree of non-free rational curves on $X$ is at least $n-2$. We
classify extremal contractions of such varieties. As an application, we obtain
a classification of Fano fourfolds with pseudoindex and Picard number greater
than one. Combining this result with previous results, we complete the
classification of smooth Fano $n$-folds with pseudoindex at least $n-2$ and
Picard number greater than one. This can be seen as a generalization of various
previous results. We also discuss the relations between pseudoindex and other
invariants of Fano varieties.
