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We introduce a class of hybrid boundary element methods for the solution of sound soft scattering problems. To facilitate the applicability of our algorithms throughout the entire frequency spectrum, we have enriched our Galerkin approximation spaces, through incorporation of oscillations in the incident field of radiation, into the algebraic and trigonometric polynomial approximation spaces. We utilize, alternatively, a change of variables around the transition regions so as to guarantee that the approximation spaces are adapted to the asymptotic behavior of solutions.The resulting methodologies have three distinctive properties. Indeed, from a theoretical point of view, they can be tuned to demand only an O(k^a) increase (for any a > 0) in the number of degrees of freedom to maintain a fixed accuracy with increasing wavenumber k. Perhaps more importantly, from a practical point of view, they give rise to linear systems with significantly enhanced condition numbers and this, in turn, allows for more accurate solutions if desired.