WTW for this part of jazz songs

I think you've already hit upon it. That's called taking a "solo", or more generally, "improvising" over the changes. In certain styles of jazz, the "solo" section tends to be fairly sparse and minimalistic, as compared to the solo section in pop, rock, or country, where one instrument wails loudly over the other instruments. Usually, during this solo section, the drums will take more liberties rhythmically, but continue playing, the lead instrument will drop out nearly completely, and the piano will only play sporadic chords, keeping the harmonic progression moving forward. This is called "comping" on the part of the pianist. Knowing how to "comp" during the solo section is considered a fundamental aspect of playing jazz as a pianist.

Sometimes, jazz performers will swap very quickly between themselves, handing over the job of improvising between members of the trio in quick succession. This is called "trading fours", because usually each performer gets four measures to improvise over the changes before handing the solo over to the next person – often by visual cues like a nod in their direction.

"Cadenza" is rarely seen in jazz contexts, as it's usually the classical term for the improvisation of the melody by a performer. It's very similar to the "solo" section in more informal genres, but in a cadenza, the orchestra usually stops entirely, and allows the performer to take massive liberties with the tempo, chords, melodies, and even key signature, for a predetermined number of measures.