An action of a semigroup \(S\) on a set \(X\) is called semi-transitive if for every pair \((x,y)\) in \(X\) we have some member \(a\) in \(S\) such that either \(x.a=y\) or \(y.a=x\); this therefore generalises the usual definition of transitivity. The purpose of this paper is to describe the structure of semitransitive subsemigroups of the finite symmetric inverse semigroup \(I_n\) of minimal cardinality up to the classification of transitive subgroups of minimal cardinality of finite symmetric groups. One theorem describes the minimal transitive subsemigroups of the subsemigroup of all maps with finite domain within an infinite symmetric inverse semigroup, which yields a similar result for minimal transitive subsemigroups of finite symmetric inverse semigroups. It is proved, for example, that the cardinality of a semitransitive but not transitive subsemigroup of \(I_n\) is at least \(n+1\) and that the subsemigroups \(S\) that have this minimal possible cardinality are certain wreath products of transitive groups and chain monoids.