Extensions to the Roe and HLL method have been previously formulated in order to solve the Shallow Water equations in the presence of source terms. These were named the Augmented Roe (ARoe) method and the HLLS method, respectively. This paper continues developing these formulations by examining how entropy corrections can be appropriately fitted in for the ARoe method and how the HLLS method can be formulated more generally. This is done in two ways. Firstly, this paper extends the reasoning of Harten and Hyman required by the ARoe method to include the source term contributions and thus arrives to a more complete formulation of the entropy fix, which will be compared with the approximation presented in previous works through numerical experiments. Secondly, it is shown how a relaxation of the criteria used when choosing waves in the HLLS method yields better solutions to problems where the HLLS would previously fail. In summary, this paper seeks to offer a comprehensible review of the ARoe and HLLS methods while improving its performance in cases with transcritical rarefactions for the inhomogeneous Shallow Water Equations in one dimension
