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Self-force methods can be applied in calculations of the scatter angle in two-body hyperbolic encounters, working order by order in the mass ratio (assumed small) but with no recourse to a weak-field approximation. This, in turn, can inform ongoing efforts to construct an accurate description of the general-relativistic binary dynamics via an effective-one-body description or other approaches. Existing self-force methods are to a large extent specialised to bound, inspiral orbits. Here we derive the first-order conservative self-force correction to the scattering angle, show its agreement with recent post-Minkowsian results, and develop a technique for (numerical) self-force calculations that can efficiently tackle scatter orbits. In the method, the metric perturbation is reconstructed from a Hertz potential that satisfies (mode-by-mode) a certain inhomogeneous version of the Teukolsky equation. The crucial ingredient in this formulation are certain jump conditions that the (multipole modes of the) Hertz potential must satisfy along the worldline of the small body's orbit. We present a closed-form expression for these jumps, for an arbitrary geodesic orbit in Schwarzschild spacetime. To begin developing the numerical infrastructure, a scalar-field evolution code on a Schwarzschild background (in 1+1D) is developed. Following this, results for the conservative scalar self-force corrections to the scatter angle are calculated. We continue by constructing a Teukolsky evolution code on a Schwarzschild background. This produces numerically unstable solutions due to unphysical homogeneous solutions of the Teukolsky equation at the horizon and null infinity being seeded by numerical error. This can be resolved by a change of variables to a Regge-Wheeler-like field. We then present a full numerical implementation of this method for circular and scatter orbits in Schwarzschild.