Movement efficiency is just a fancy way of saying that an athlete is well coordinated. The best athletes use the fire their muscles in the right sequence and with appropriate levels of force.
Good example will be high jumper, who has great movement efficiency. They fire their muscles in the right order, they move limbs synergistically and as a result, they jump really high (and he make it look easy).
Movement efficiency, during the jump, allows an athlete to generate the bridge amount of power with the least amount of effort.
We must remember that different sport disciplines have different technique of jumps and also need other movement efficiency. Volleyball players and athletes training just to do well in the standing jump test won’t need to spend as much time working on movement efficiency as athletes wanting to perform a 360 degree slam dunk or compete in track and field events.
Of the 4 main jumping attributes movement efficiency is by far the least complicated to develop. However just because it isn’t complicated doesn’t mean it is always easy either!
All you really need to do to improve your jumping movement efficiency is to go and practice jumping in the manner in which you use it for your sport. If you want to improve your jumping efficiency for volleyball then you must spend some additional time for practice jumping for maximal height in the manner used in volleyball (spikes, blocks, services).
Remember that no program, coach, facilities, whatever, do not help you if you do not go to the hall and do not do a few jumps!
Aside from improving movement efficiency, practicing jumping also develops jump specific speed and strength as good if not better than basically any drill or exercise you can think of. Many people overlook this key principle because it just seems too simple.
There is no doubting that weight training and plyometric/jumping drills help develop strength and speed for improving your jump.
So why if jumping practice improves movement efficiency and jump related speed and strength better than anything else do you even bother with other drills and exercises? This is the obvious question to ask and it boils down to the fact that after a while your gains stagnate and you need to find other ways to increase your ability to generate jumping power.