A second-order accurate augmented method is proposed and analyzed in this paper for a general elliptic PDE with a general boundary condition using the jump embedded boundary conditions (JEBC) formulation. First of all, the existence and uniqueness of an interface problem with given are discussed. Then, the well-posedness theory is extended to the interface problems with given jump conditions. In the proposed numerical method, one novel idea is to preconditioning the PDE first so that the coefficient of the highest derivative is of $O(1)$. The second idea is to introduce two augmented variables corresponding to the jump in the solution and its normal derivative along the boundary to get an interface problem. For a piecewise constant coefficient, the fast Poisson solver then can be utilized in a rectangular domain. The augmented variables can be determined from a Schur complement system. We also propose two preconditioning techniques for the GMRES iterative method for the Schur complement; one is from the flux jump condition, and the other one is from the algebraic preconditioner based on the interpolation scheme in the augmented algorithm. The presented numerical results show that the proposed method has not only obtained second order accurate solutions in the $L^∞$ norm globally, but also second order accurate normal derivatives at the boundary from each side of the interface. The proposed preconditioning technique can speed up 50-90% compared with the method without preconditioning.