We consider holomorphic maps f:U→U for a hyperbolic domain U in the complex plane, such that the iterates of f converge to a boundary point ζ of U. By a previous result of the authors, for such maps there exist nice absorbing domains W⊂U. In this paper we show that W can be chosen to be simply connected, if f has doubly parabolic type in the sense of the Baker-Pommerenke-Cowen classification of its lift by a universal covering (and ζ is not an isolated boundary point of U). We also provide counterexamples for other types of the map f and give an exact characterization of doubly parabolic type in terms of the dynamical behaviour of f