In [An adaptive finite volume solver for steady Euler equations with non-oscillatory k-exact reconstruction, G. H. Hu and N. Y. Yi, J. Comput. Phys., Vol. 312, pp. 235–251], a high order finite volume method was proposed for steady Euler equations. The method consists of a Newton iteration method to linearize the governing equations, and a geometrical multigrid method to solve the derived system. The non-oscillatory k-exact reconstruction was developed towards the high quality solution reconstruction, and h-adaptive method was introduced to handle the efficiency issue. To further improve the numerical method, we study the following two issues in detail in this paper. First, we test the new reconstruction patch for the element which locates on the domain boundary, which could improve the convergence of the steady state significantly. Second, the adjoint-based a posteriori error estimation is introduced for the h-adaptive method, which could deliver more efficient adaptive refinement of the mesh. The numerical tests show the effectiveness of the proposed method.