Here’s that rainy day is not only a good tune, but a tune that is really fun to play as it has some pretty interesting changes. 
The tune is normally played in G, but can be played in any key. Bill Evans starts out in B and there is a modulation in the ending of the tune.

So, what can we do to make this tune interesting? We can do a lot!
I created a 10 step process, a 10 step tutorial on what I think we can do. (Off course, this was just a selection of the endless possibilities) 

Anyway
Have a look at my YouTube video here:

This is how jazz standards are played normally. You play the melody simple first before you make it more freely.
All of this comes from the idea that jazz should be “grounded” in the roots before you develop it to something more interesting. If we are going to make some changes, we need to set a foundation, something we can change from. 

My favorite recording of this is (as you might have noticed): Bill Evans recording from his Alone album. Verve Records 1968.
So: Listen a lot to your favorite record to get as influenced as possible. Just make sure you do step 3 very well and you’ll be fine!

This means you add whatever you like to the music. And off course, this goes for any jazz tunes – jazz standards or your own originals.

I always say that to play jazz is to constantly look for ways to tweak the core elements. So you look for ways to do things differently than before.

Pick stuff from your bag of tricks that you can add to the song (here’s that rainy day or any other jazz standard)

Re-arrange the music as well. Add a few bars, add a bar in 5 or 6 or any other idea you can give to the music

Keep it simple or complex depending on what you feel is the right for the music
Sometimes I feel for making a simple improvisation that is pretty close to the original melody. Other times I want to make it more complex.
Keit Jarrett talks about finding the music or not. When I recorded this, I had just a “directional” idea of how it would turn out. I had no clue that it would turn out the way it did.
So when I recorded this, I simply closed my eyes and listened.

Too often, we as teachers tend to talk about the technical stuff (The left-brain stuff).

However: Music requires you to get in touch with your right brain (or heart?), emotions etc.

Connect the dots, play the melody again but add some difference to what you started with.
Last time you play the melody, try to play it somewhat different than you did in the beginning.

In the end of a ballad like this, I really like to slow things down. This is not only a Ritardando, but you stop moving that much. (Don’t play so much of the eight notes between the melody notes)

Why not end fresh? End the tune with something more interesting than just end it on the Tonic.