I think Cicci did "move up" somewhat during the film. When we first see him, he's in the background when Michael and Sonny are saying goodbye, then around the piano during the Mob war, then crowd control when Sonny beats up Carlo. But by the end of the film, he seems to have gained some importance when he tells Tom and Tessio that "the boss says for you to go on ahead..." And, as we saw in II, he's Pentangeli's right-hand man.