In this paper, an adjoint-based h-adaptive high-order reconstructed DG (rDG) method is introduced for solving the two dimensional steady-state compressible Euler equations. Based on the hybrid reconstruction strategy developed in [9, 28], adjoint-based a posteriori error estimation is further derived and developed for h-adaption. The formulation of error indicator is carefully investigated in order to deliver better approximation with respect to the error in the computed output functional. In order to evaluate the performance of the proposed method, an adjoint-based h-adaptive rDG(p₁ p₂) method is implemented, in which a hybrid p₁ p₂ reconstruction and a hybrid p₂ p₃ reconstruction are adopted in the primal solver and the adjoint solver to obtain the primal solution and the adjoint solution, respectively. A number of typical test cases are selected to assess the performance of the adjoint-based h-adaptive hybrid rDG method. The hybrid reconstruction strategy combined with h-adaptive techniques based on adjoint-based error estimate presented in this work demonstrates its capacity in reducing the error with respect to the computed output functional and improving the level of accuracy for numerical simulations of the compressible inviscid flows.