Here are some passing chord ideas for “Fly Me To The Moon” that will take you past what’s in the Real Book. I like to use altered 7th chords a half step above whatever chord I’m moving to. There is also a little diminished and augmented thing that works well on this tune.
Stacked Fourth Chords
I like to add stacked fourths to my playing sometimes. These little stacked fourth chords can be a cool way to spice up or comping, and you can also throw them in while you are improvising.

 
			