We introduce the intermediate set as an interpolating solution concept between the core and the Weber set of a coalitional game. The new solution is defined as the limiting superdifferential of the Lovász extension and thus it completes the hierarchy of variational objects used to represent the core (Fréchet superdifferential) and the Weber set (Clarke superdifferential). It is shown that the intermediate set is a non-convex solution containing the Pareto optimal payoff vectors that depend on some chain of coalitions and marginal coalitional contributions with respect to the chain. A detailed comparison between the intermediate set and other set-valued solutions is provided. We compute the exact form of intermediate set for all games and provide its simplified characterization for the simple games and the glove game.