Transpose in various keys is an important verification of having learned a standard. This practice helps us to overcome the technical and spatial references of our instrument. This lead us to think in terms of sounds.
Going deeper and deeper into a jazz standard we can analyze the chord changes and then translate into harmonic degrees of the scale the structure of the song, note the points where it modulates and the interval between the last chord and the first of the new key.
We can now begin to apply the degree scheme and modulation intervals to another key of our song.
These references needs to be memorized, mentally we must make “to play”chords in our “ear” as they flow. Degrees and intervals mean sounds and them relationships.
Even the melody of the theme can be rationalized, especially in particular points. It should be enough focus on the interval of the starting note of the melody respect to the chord and the knowledge of the melody will automatically make us sing/playing it in the new key.