Type-directed component-based program synthesis is the task of automatically building a function with applications of available components and whose type matches a given goal type. Existing approaches to component-based synthesis, based on classical proof search, cannot deal with large sets of components. Recently, Hoogle+, a component-based synthesizer for Haskell, overcomes this issue by reducing the search problem to a Petri-net reachability problem. However, Hoogle+ cannot synthesize constants nor $\lambda$-abstractions, which limits the problems that it can solve. We present Hoogle$\star$, an extension to Hoogle+ that brings constants and $\lambda$-abstractions to the search space, in two independent steps. First, we introduce the notion of wildcard component, a component that matches all types. This enables the algorithm to produce \textit{incomplete functions}, i.e., functions containing occurrences of the wildcard component. Second, we complete those functions, by replacing each occurrence with constants or custom-defined $\lambda$-abstractions. We have chosen to find constants by means of an inference algorithm: we present a new unification algorithm based on symbolic execution that uses the input-output examples supplied by the user to compute substitutions for the occurrences of the wildcard. When compared to Hoogle+, Hoogle$\star$ can solve more kinds of problems, especially problems that require the generation of constants and $\lambda$-abstractions, without performance degradation.