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The simplex method for linear programming is known to be highly efficient in practice, and understanding its performance from a theoretical perspective is an active research topic. The framework of smoothed analysis, first introduced by Spielman and Teng (JACM '04) for this purpose, defines the smoothed complexity of solving a linear program with \(d\) variables and \(n\) constraints as the expected running time when Gaussian noise of variance \(\sigma^2\) is added to the LP data. We prove that the smoothed complexity of the simplex method is \(O(\sigma^{-3/2} d^{13/4}\log^{7/4} n)\), improving the dependence on \(1/\sigma\) compared to the previous bound of \(O(\sigma^{-2} d^2\sqrt{\log n})\). We accomplish this through a new analysis of the \emph{shadow bound}, key to earlier analyses as well. Illustrating the power of our new method, we use our method to prove a nearly tight upper bound on the smoothed complexity of two-dimensional polygons. We also establish the first non-trivial lower bound on the smoothed complexity of the simplex method, proving that the \emph{shadow vertex simplex method} requires at least \(\Omega \Big(\min \big(\sigma^{-1/2} d^{-1/2}\log^{-1/4} d,2^d \big) \Big)\) pivot steps with high probability. A key part of our analysis is a new variation on the extended formulation for the regular \(2^k\)-gon. We end with a numerical experiment that suggests this analysis could be further improved.