We propose a new method for investigating the global properties of the retarded Green's function G R(x′,x) for fields propagating on an arbitrary globally hyperbolic spacetime. Our method combines the Hadamard form for G R (this form is only valid within a normal neighborhood of x) together with Kirchhoff's integral representation for the field in order to calculate G R outside the maximal normal neighborhood of x. As an example, we apply this method to the case of a scalar field on a black hole toy-model spacetime, the Plebański-Hacyan spacetime, M 2×S2. The method allows us to determine in an exact manner that the singularity structure of the "direct" term in the Hadamard form for G R(x′,x) changes from a form δ(σ) to "1/πσ" after the null geodesic joining x and x′ has crossed a caustic point, where σ is the world function. Furthermore, there is a change of form from a Heaviside step function H(-σ) to a "-ln|σ|/π" in the "tail" term, which has not been explicitly noted before in the literature. We complement the results from the Kirchhoff integral method with an analysis for large- l of the Green's function modes. This analysis allows us to determine the singularity structure after null geodesics have crossed an arbitrary number of caustics, although it raises a causality issue which the Kirchhoff integral method resolves. Because of the similarity in the caustic structure of the spacetimes, we expect our main results for wave propagation to be valid also on Schwarzschild spacetime. © 2012 American Physical Society.