Dynamics of a one-parameter family of functions \(f_\lambda (z)=\lambda + z+\tan z, z \in {\mathbb{C}}\) and \(\lambda \in {\mathbb{C}}\) is investigated in this article. This function \(f_\lambda\) has an unbounded set of singular values. The dynamics of \(f_{\lambda +m\pi }\) is determined for all non-zero integers m and for the values of \(\lambda\) satisfying either \(|2+\lambda ^2|<1,\) or \(\lambda =i,\) or \(2+\lambda ^2=e^{2\pi i \alpha }\) for all rational numbers \(\alpha\) and for each bounded type irrational number \(\alpha .\) For such values of \(\lambda ,\) the existence of |m| many wandering domains of \(f_{\lambda +m\pi }\) with disjoint grand orbits contained in the lower half-plane is asserted along with a completely invariant Baker domain containing the upper half-plane. Further, each such wandering domain is found to be simply connected, unbounded, and escaping. Different types of the internal behavior of \(\{f^n_{\lambda +m\pi }\}_{n>0}\) on such a wandering domain W are highlighted for different values of \(\lambda .\) More precisely, for \(\mid 2+\lambda ^2\mid <1,\) it is shown that the forward orbit of every point \(z\in W\) stays away from the boundaries of \(W_n\) s. For \(\lambda =i,\) it is proved that \(\lim _{n\rightarrow \infty }dist(f^n_{i+m\pi }(z),\partial W_n)=0\) for each \(z\in W.\) Further, \(\Im (f^n_{i+m\pi }(z))\rightarrow -\infty\) as \(n \rightarrow \infty .\) For \(2+\lambda ^2=e^{2\pi i\alpha }\) for each rational number \(\alpha ,\) \(\lim _{n\rightarrow \infty }dist(f^n_{\lambda +m\pi }(z),\partial W_n)=0\) is established for each \(z\in W.\) But, \(\Im (f^n_{\lambda +m\pi }(z))\) tends to a finite point for each \(z\in W\) whenever \(n \rightarrow \infty .\) For \(2+\lambda ^2=e^{2\pi i\alpha },\) \(\lim _{n\rightarrow \infty }dist(f^n_{\lambda +m\pi }(z),\partial W_n)>0\) for each \(z\in W\) and for each bounded type irrational number \(\alpha .\)
