A computational procedure that allows the detection of a new type of high-dimensional chaotic saddle in Hamiltonian systems with three degrees of freedom is presented. The chaotic saddle is associated with a so-called normally hyperbolic invariant manifold (NHIM). The procedure allows us to compute appropriate homoclinic orbits to the NHIM from which we can infer the existence of a chaotic saddle. It also allows us to detect heteroclinic connections between different NHIMs. NHIMs control the phase space transport across an equilibrium point of saddle-centre- -centre stability type, which is a fundamental mechanism for chemical reactions, capture and escape, scattering, and, more generally, 'transformation' in many different areas of physics. Consequently, the presented methods and results are of broad interest. The procedure is illustrated for the spatial Hill's problem which is a well-known model in celestial mechanics and which gained much interest, e.g. in the study of the formation of binaries in the Kuiper belt.