You build an orbital ring above the altitude of the LEO satellites, use non-synchronous skyhooks to pick up payloads from inside the atmosphere at an altitude that can be efficiently served by aeroplanes, and then fling them the rest of the way to geosynchronous or interplanetary transfer orbits by accelerating them along the ring.
https://arxiv.org/abs/1505.07424
You build an orbital ring above the altitude of the LEO satellites, use non-synchronous skyhooks to pick up payloads from inside the atmosphere at an altitude that can be efficiently served by aeroplanes, and then fling them the rest of the way to geosynchronous or interplanetary transfer orbits by accelerating them along the ring.