  For a presentation $\mathcal{A}$ of a transversal matroid $M$, we study the
set $T_{\mathcal{A}}$ of single-element transversal extensions of $M$ that have
presentations that extend $\mathcal{A}$; we order these extensions by the weak
order. We show that $T_{\mathcal{A}}$ is a distributive lattice, and that each
finite distributive lattice is isomorphic to $T_{\mathcal{A}}$ for some
presentation $\mathcal{A}$ of some transversal matroid $M$. We show that
$T_{\mathcal{A}}\cap T_{\mathcal{B}}$, for any two presentations $\mathcal{A}$
and $\mathcal{B}$ of $M$, is a sublattice of both $T_{\mathcal{A}}$ and
$T_{\mathcal{B}}$. We prove sharp upper bounds on $|T_{\mathcal{A}}|$ for
presentations $\mathcal{A}$ of rank less than $r(M)$ in the order on
presentations; we also give a sharp upper bound on $|T_{\mathcal{A}}\cap
T_{\mathcal{B}}|$. The main tool we introduce to study $T_{\mathcal{A}}$ is the
lattice $L_{\mathcal{A}}$ of closed sets of a certain closure operator on the
lattice of subsets of $\{1,2,\ldots,r(M)\}$.
