This paper continues the study of polynomial hulls by the authors. Let Y be a compact subset of \({\mathbb{C}}^ 2\) contained in \(\partial \Delta \times \Delta\), where \(\Delta\) is the open unit disk in the plane such that for \(\lambda\in \partial \Delta\) the complement of \(Y_{\lambda}:=\{w:\) (\(\lambda\),w)\(\in Y\}\) is connected. The polynomial hull of Y is denoted by \(\hat Y.\) For a certain class \({\mathcal F}\) of bounded and holomorphic functions on \(\Omega_ 0:=\Delta \times ({\mathbb{C}}\setminus \Delta)\), Theorem 1 proves the existence of holomorphic extensions to \(\Omega^*:=(\Delta \times {\mathbb{C}})\setminus \hat Y\). Theorem 2 says that if \(Y_{\lambda}\) is convex for \(\lambda\in \partial \Delta\), then the extensions of particular functions in \({\mathcal F}\) have no zeros on \(\Omega^*\).