Granted that you have an understanding of the harmonic make up of the progression, the best way to memorize a tune is to break it down into several cadences. Determine its structure (eg. AABA, AAB, ABC, etc) and then learn one section at a time. You can even learn each section in 4 measure chunks.
Take “Satin Doll” for example. This is a 32 measure AABA form. In the key of C the 1st A section (or 8 measures) is made up of:
IIm7 – V7 (Dm7 – G7)
IIm7 – V7 (Dm7 – G7)
IIm7 -V7 /IIm7 (secondary II-V of the II) (Em7 – A7)
IIm7 -V7 /IIm7 (secondary II-V of the II) (Em7 – A7)
IIm7 -V7 /V7 (secondary II-V of the V) (Am7 – D7)
sub IIm7 -V7 /Imaj7 (substitute II-V of the I) (Abm7 – Db7)
Imaj7 (Cmaj7)
IIm7b5 -V7alt /IIm7 (secondary II-V of the II) (Em7b5 – A7b9)
Understanding the harmonic components, enables us to learn the tune in short familiar chunks which in turn reveal how we should improvise each phrase. That’s why it is crucial that we first perform a harmonic analysis of the tune.
Hope this helps!
Richie