It is typical of its time, comprising two 16-bar themes in one key followed by a more leisurely 32-bar theme in which we modulate into the key a fifth below. This final theme was called the 'Trio' - a term whose usage dates at least from the classical music of the 18th Century.
Think of At a Georgia Camp Meeting, Climax Rag, Hiawatha Rag and Buddy's Habit. They are constructed in a similar way.
Our jazz band version of Blame It On The Blues is remarkably faithful to the original sheet music, (using Lasse's labels) in Themes A and B. But what we play as Theme C (the Trio) is a simplification and reinterpretation of the notes Cooke wrote for the piano. Here's his original Theme C (The Trio). Note that it also had a 4-bar Bridge which our bands do not play: