Well nothing is free. It takes work and attention to detail to move as efficiently and effectively as possible. It's all about time and space.
The technical side has to do with first steps, drops, crossovers etc. It also has to include getting to the ball, behind it, early enough so that your last step isn't used to get to the ball, which usually results in complete closing of stance, blocking of hips, and prevention of the rotational and/or linear transfer used as part of hitting the ball in perfect, balance a book on the head, balance. Using the last step to get to the ball usually includes two or more additional steps to recover one's balance. Those extra steps to make up for a poor set up result in a waste of time and an increase in the distance you cover to hit the ball and which you must cover again to regain the center of possible returns. For the technical "choreography" of tennis you would be best served watching video of the best smooth movers among the pros and listening to the pro you've entrusted your improvement to.
The pure technical aside, things you can do right now to improve court coverage, reaction, anticipation, movement and recovery are:
1) tune into the ball sooner-see it off the opponent's racket. You may think you are but if you examine it you will find that, like most people struggling with footwork, you are probably not aware of where the ball is going until it is crossing the net on its flight toward you.
2) tune into where the ball will bounce on your side of the court. Most players are fairly proficient at judging pace and lateral direction, but most are almost totally unaware of the depth at which the opponent's shot will bounce. Explore it. Tune into it and almost magically you will find yourself moving to the ball on the diagonal, rather than first running parallel to the baseline and then changing direction and moving toward the ball. Sounds simple but I'll bet a paycheck that you do the latter.
3) strive to be behind the ball-not merely getting to the ball-as the ball bounces on your side of the court. This will quicken you up and you will give yourself a better opportunity to hit from a balanced position.
4) stress BALANCE. Become conscious of how your body is balanced during every step and every movement on the court. Explore your balance, be much more aware of it and I mean AWARE OF IT. In the ready, while moving toward the ball, while hitting the ball and while recovering the center.
5) be smooth. A tell tale sign of inefficient movement is bouncing vision. Jarring, overly bouncy strides are a sure sign of reactionary movement. A clear, non-jittery view during play is what to strive for.
6) count your steps after a stroke. If you take 2 or 3 steps after a shot, you were off balance and probably late getting to it, which takes you right back to #'s 1 and 2 on this list.
Cues:
-see the hit
-anticipate and be aware of the bounce tune into it
-Explore your balance. Be in balance 1000% of your time on court, from your serve to turning and running down a lob over your head, every single movement.
-a smooth view indicates you're moving as efficiently as possible
Being more keenly aware of these elements are immediate movement enhancers. The ability to maintain that awareness on the practice court and in matches and to make it part of your game without thinking about it are yet other "learned" skills for most players.
Good luck,
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