The authors continue their recent work on common hypercyclic vectors for paths and similarity orbits of operators \(T\) on a separable Hilbert space \(H\) [J. Oper. Theory 61, 191--233 (2009); ibid. 66, No.~1, 107--124 (2011; Zbl 1230.47021); J. Math. Anal. Appl. 375, No.~1, 139--148 (2011; Zbl 1208.47013); Integral Equations Oper. Theory 65, No.~1, 131--149 (2009; Zbl 1196.47008)]. In the present paper, they investigate the unitary orbit of an operator \(T\), that is, the set of all operators of the form \(U^* T U\), with \(U\) a unitary operator on \(H\). Since all the elements of the unitary orbit have the same norm of \(T\), the orbit cannot be norm dense in the space of all operators. The unitary orbit is path connected. The authors prove that, if \(T\) is a hypercyclic operator on \(H\), then its unitary orbit contains a path of operators whose closure in the strong operator topology contains the orbit and the whole path has a \(G_\delta\) set of common hypercyclic vectors. Several consequences and improvements of this result are also included.