For low energy spacecraft trajectories such as multi-moon orbiters for the Jupiter system, multiple gravity assists by moons could be used in conjunction with ballistic capture to drastically decrease fuel usage. In this paper, we investigate a special class of multiple gravity assists which can occur outside of the perturbing body's sphere of influence (the Hill sphere) and which is dynamically connected to orbits that get captured by the perturber and orbits which escape to infinity. We proceed by deriving a family of symplectic twist maps to approximate a particle's motion in the planar circular restricted three-body problem. The maps capture well the dynamics of the full equations of motion; the phase space contains a connected chaotic zone where intersections between unstable resonant orbit manifolds provide the template for lanes of fast migration between orbits of different semimajor axes. Within the chaotic zone, the concept of a set of reachable orbits is useful. This set can be considered bounded by, on one end, orbits leading to ballistic capture around the perturber, and on the other end, the orbits escaping to infinity or a bounding surface at finite distance.