This paper considers semi-global stability of an observer-based extremum-seeking (ES) scheme acting on a Hammerstein plant. Unlike previous semi-global analyses, no restriction is placed on the speed of the ES dynamics relative to the plant dynamics. For any set of frequencies of a probing sinusoidal "dither," it is shown how the plant output can be made to converge to an arbitrarily small neighborhood of its minimum from an arbitrarily large set of initial conditions. Furthermore, in the absence of noise, it is shown how increasing the dither frequency can allow arbitrarily fast convergence of the plant input to a small neighborhood of its optimum. Practical application of the results requires no a priori knowledge of the plant nonlinearity; however, in order to design the ES scheme to operate over a wide range of dither frequencies, some knowledge of the plant dynamics is required. Simulation examples are used to demonstrate these results and to highlight practical considerations when using a high-frequency dither.