The main obstacle to incremental sentence processing arises from right-branching constituent structures, which are present in the majority of English sentences, as well as from optional constituents that adjoin on the right, such as right adjuncts and right conjuncts. In CCG, many right-branching derivations can be replaced by semantically equivalent leftbranching incremental derivations. The problem of right-adjunction is more resistant to solution, and has been tackled in the past using revealing-based approaches that often rely either on the higher-order unification over lambda terms (Pareschi and Steedman, 1987) or heuristics over dependency representations that do not cover the whole CCGbank (Ambati et al., 2015). We propose a new incremental parsing algorithm for CCG following the same revealing tradition of work but having a purely syntactic approach that does not depend on access to a distinct level of semantic representation. This algorithm can cover the whole CCGbank, with greater incrementality and accuracy than previous proposals.