The analytic continuation of the GW self-energy from the imaginary to the real energy axis is a central difficulty for approaches exploiting the favourable properties of response functions at imaginary frequencies. Within a scheme merging contour-deformation and analytic-continuation techniques, we show on the basis of extensive calculations for large molecular sets that it is preferable to perform an analytic continuation of the dynamically screened Coulomb potential W rather than the much more structured self-energy operator. The case of states lying far away from the gap, including core states, is addressed by generalizing the analytic continuation scheme, accounting further for quasiparticle lifetimes.