This is the old way to think about things. CaptureRail is a better way to use launch loop technologies to "land" payloads at orbiting destinations.
A rotating tether in an elliptical orbit below GEO may have a perigee of 16881 km, an apogee of 42164 km, and an apogee center-of-mass velocity of 2325 m/s . If it is 60 kilometers long, and rotating once per 250 seconds, the tip velocity is 750 m/s and the centrifugal acceleration is 9.4 m/s. That can snatch a vehicle at the apogee of a 1575 m/s transfer orbit, and release it at 3075 m/s 125 seconds later, at geosynchronous altitude and velocity. Additional momentum must be added back to the tether, either be done by capturing something from GEO and sending it back downwards, or by adding delta-V with fuel-efficient ion engines (with exhaust velocities so high that the ejecta gets thrown out of the solar system!). This will provide the quickest trip to GEO, but the latency between launch windows will be long.
The rotating tether apogee will "surface" at different places along the equatorial belt at different and perhaps inconvenient times. For frequent trips to popular points, a geostationary "elevator" tether might be preferable. This might extend from 29000 km radius (0.7 times GEO radius ) to 51000 km radius (1.2 times GEO radius). Vehicles rendezvous with the bottom end, and are pulled up to GEO, or even beyond, to be released into transfer orbits reaching the moon. Extra momentum can be supplied by vehicles in slingshot orbits around the moon, attaching at higher altitudes and climbing down(1). The trip will be longer - the transfer orbit to 29000 km is only 3.4 hours, but the climb up the tether will take 8.5 hours at 400 meters per second elevator speeds.
While orbit management and rendezvous will be tricky (you wait a long time if you miss!), with good orbital calculations and precision-timed releases from the launch loop, these rendezvous can be reliably performed. In the rare instances where we miss, the vehicle can re-enter and try again, or rendezvous with a different elevator or rotating tether, located one orbit away. Climbing that tether to the appropriate release point, we can launch into a higher transfer orbit which will intercept the desired tether sooner.
Other possibilities involve additional elevator tethers in MEO, with centers of mass in 12 hour orbits. Vehicles can climb these partway and hold, then release at the appropriate time to reach many different geostationary tethers.
(1) Note to Bill Bryson - Climbing down is normally a contradiction (see Bryson's Dictionary of Difficult Words, page 39). However, in this case, where we must forcefully pull ourselves towards the Earth, it is correct. It is also often correct underwater, where we are pulling ourselves downwards against buoyancy (see page 31).