Low Earth Orbit is a bad place for an elevator. The satellites and dense debris make an impact likely. The tether would probably get cut too often to confer a return exceeding investment.
A TALE OF TWO MOONS
A series of orbital elevators might be worthwhile. Here is a diagram of possible elevators from the Martian moons Phobos and Deimos:
Tossing a payload from the top of the 937 km Phobos tether will toss it to an ellipse whose apoaerion coincides with the foot of the 2942 km Deimos tether. Speed of the payload at apoaerion matches the speed of the Deimos elevator foot.
Going the other direction, dropping a payload from the Deimos elevator foot will send a payload to the top of the Phobos elevator. Again, the ellipse's periaerion speed matches the speed of the top of Phobos' elevator.
Thus the two moons could exchange passengers and cargo using virtually no propellent.
EARTH ORBITAL ELEVATORS
Could we make a similar relay in earth's neighborhood?
There are a few problems. With Phobos' 1.07e16 kilogram mass and Deimos' 1.5e15 kilogram mass, the moons are huge momentum banks. They could catch and throw payloads for millennia with very little effect on their orbits.
Earth elevators wouldn't enjoy such a large anchor mass. If a 20 tonne elevator imparts 1 km/s delta V to a 10 tonne payload, the elevator will suffer a velocity change of .5 km/s.
To have tethers whose orbits aren't destroyed by a catch or a throw, it's desirable to have an anchor mass that greatly exceeds the payload mass.
I've envisioned 3 vertical tethers in earth's neighborhood:
For the Super Geosynchronous Tether there is a possible source of anchor mass: The dead sats in a graveyard orbit just above geosynchronous orbit. For the other tethers it would be very difficult to get a substantial anchor mass.
If the elevator is massive enough that a few catches or throws don't destroy its orbit, it can be a momentum exchange broker. Catching payloads from above or dropping payloads to lower orbits will boost momentum. Catching payloads from below or throwing payloads to higher orbits will sap momentum. By balancing momentum boosting actions with momentum sapping maneuvers, the elevator orbit can be preserved without use of propellent.
Or if momentum does need to be changed with the use of propellent, ion engines such as Hall thrusters are an option. A xenon exhaust velocity of 30 km/s is nearly 10 times as good as chemical propellents. And if elevator catches and throws are separated by weeks or months, there is time for the low thrust ion engines to build up momentum. Elevators are a way to have you Ion engine ISP and eat Oberth cake too.
I believe orbital elevators are doable and worthwhile. They would take substantial initial investment, though.