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Source. The augmentation step in the proof of Theorem 5, printed p. 1326 (published PDF), where it is cited to Mirsky–Perfect.

Statement. Let $(C_\ell)_{\ell\in L}$ be a finite indexed family that has a full transversal. Every partial-transversal range $P$ is contained in the range of some full transversal.

Proof. Partial-transversal ranges are the independent sets of the transversal matroid, by the finite transversal-matroid theorem. Because a full transversal exists, this matroid has rank $|L|$. By finite basis extension, $P$ is contained in some base $Q$.

The base has $|Q|=|L|$ and is itself a partial-transversal range. Any witness assigns its $|L|$ distinct elements injectively to a subset of the $|L|$ family indices, so it uses every index and is a full transversal. Thus $P\subseteq Q$ has the required form. The witnessing assignment for $Q$ may rematch elements of $P$; the asserted set inclusion is preserved. For an empty family, $P=Q=\varnothing$. $\square$

The finite basis-extension input is proved in elementary finite matroid facts.