The original E-CUSP scheme suggested by Zha and Hu is modified to remove the temperature oscillation with low numerical dissipation. The pressure term in the energy equation dissipation is replaced by the total enthalpy. The scheme has low diffusion to accurately resolve wall boundary layers and to capture crisp shock waves. The accuracy of the scheme is compared with other popularly used upwind schemes. For a supersonic flat plate laminar flow, the wall temperature is precisely predicted by using the modified scheme with first order accuracy on a very coarse mesh. The scheme is also tested for the 1D Sod shock tube problem and a transonic nozzle with oblique shock waves and reflections that do not align with the mesh lines. The test cases show that the modified scheme is accurate, robust and effcient.