Interesting thought. Note that getting off 300 km up would not put you in orbit; you would lack the orbital velocity required for that. So if you get off there you'd have to burn around 8 km/s of delta-v, which is quite a bit of fuel to haul up.

For efficiency you'd want to go higher. There'd be a point much higher than 300 km where detethering would put you into a highly elliptical orbit where the periapsis is 300 km. Then you'd just have to burn retrograde to circularize. Off hand I'm not sure what the math for that would be but it should be pretty simple algebra.

Actually the cable will extend well beoynd that point. 36Mm is the altitide where the elevator would be travelig at orbital speed. Jump out any lower and you will drop back to earth. Unless you jump very hard (using rockets or something).