This paper investigates the orders of close-to-convexity for analytic functions of complex order defined in the open unit disc. Sufficient conditions are established under which certain integral operators transform analytic functions into the class κ(γ,δ), representing a generalized form of the classical close-to-convex function class. Several new extensions of previously established results are obtained, together with sharp bounds for the related differential inequalities. The study further enhances the understanding of the geometric behavior of analytic functions under the action of complex-order integral operators.