Things get much easier if a space elevator cable does not reach the ground, then it can be constructed with current materials. This is the Skyhook concept. A supersonic vehicle (wholly or partly air breathing) would connect with a tether hanging down from a satellite (connecting at a speed of Mach 10 or higher? It’s relative velocity that matters, and in flight refuelling at high speed is well proven).
The two options are a fixed tether ( basically, a short space elevator) along which the space ship would crawl up (how? Linear motors?). The alternative is a rotating tether, which would slingshot the spaceship into orbit, as shown in the diagram below:
Studies by Boeing suggest that a rotating tether is more efficient than a fixed one, but this technology allows one to start small and then work out which is best in practice. With a rotating tether, Newton’s third law still applies, and the reaction to the slingshot would lower the anchoring satellite, which would need to lifted back up, possibly through electrodynamics.
One could start connecting with a launch vehicle at a relatively high level and a connecting speed of Mach 10 or higher, then with experience and developing skyhook materials, making the cable longer and connecting at lower altitude and at lower speeds; Mach 5 or lower could be achieved with airbreathing scramjets, at much lower cost than with rockets. Eventually and gradually, learning as one goes along, skyhooks may evolve into fully fledged space elevators.
Alternatives to Rockets: Space Elevators and Launch Loops