  In our previous work, for a given Riemann surface $Y_0$ with marked handle,
we investigated geometric properties of the set of marked once-holed tori $X$
allowing holomorphic mappings of $X$ into $Y_0$. It turned out that it is a
closed domain with Lipschitz boundary. In the present paper we show that the
boundary is never smooth. Also, we evaluate the critical extremal length for
the existence of holomorphic mappings in terms of hyperbolic lengths.
