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At an altitude lower than geostationary orbit altitude, ~36000 kms, a craft on the elevator will be travelling slower than the velocity needed for a circular orbit and therefore would fall back to Earth if it let go of the elevator below a certain height (probably about ~15000kms, just a guess).
If it let go above that height but below ~36000kms it would be in orbit with its apoapsis at the point where it let go and its periapsis at an altitude somewhere between the surface of the Earth and the altitude where it let go.
A craft on the elevator above ~36000kms would be travelling faster than the velocity needed for a circular orbit so if it 'let go' of the elevator it would find itself in an orbit with the periapsis at the point where it let go and an apoapsis higher than that, depending on how high, and therefore how fast, it was.
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