This post is for anyone who is intimidated by tunes with difficult or fast moving chord progressions. I chose Serenity as an example tune because, while it's nowhere as hard as a tune like Countdown, it's still has some unique harmony that can present a challenge. I wrote the etude below as an illustration of where to start with a difficult tune, and I'll be referencing measure numbers of the etude in my tips The links below it are to high quality PDFs for tenor and alto and two recordings of the etude in varying tempos.
Outline the Basic Harmony - One of the first things you want to do be able to do on a new tune, difficult or not, is outline the harmony clearly. You'll need to figure out the exact modes and chords that the melody implies, and then make sure you can cleanly and clearly make your way through the changes even if it's just playing a bass line through the changes. If you notice my etude doesn't step out into any harmonically advanced territory until the 5th measure and even then it's just a #11 which is pretty standard on dominant chords. Some of you might be looking at measure 2 and wondering how that's basic harmony, but if you look at actual melody of Serenity you'll see that the F (the #5 or b13) is being held for over half the measure bringing me to my next point...
Use the Melody as Your Guide - The melody is your best reference for how to treat the tricky parts of the tune. In measure 3 of the melody Joe Henderson basically guides the melody in a pattern down the #11ths of the BbMaj7(#11) and AbMaj7(#11). I took my cue from him and did a similar pattern descended down the major 7ths. Another good example is on C7(#9) in measure 12. I used the exact same notes from Joe Henderson's melody, just in a different order and shape.
Use Good Voice Leading - Using good voice leading will help your improvisation sound smooth and melodic. Angular improvisation can sound great, but being able to first transition between tonalities smoothly by step will help you hear the harmony more clearly. I tried to use mainly smooth step transitions between tonalities in my etude like half step motion moving from the B7 to the Gmin7 in measures 4 and 5 or moving from the AbMaj7 to the G7 in measures 9 and 10.
Memorize the Changes - Memorizing the chord progression will allow you to hear the progression more intuitively. Knowing what the next twist in the progression is before it happens will help you transition more smoothly into the new tonality.
Repetition, Repetition, Repetition - Sometimes there are no shortcuts. Repetition is often the key to learning new tunes. Practice improvising over and over again.