The numerical approximation of Helmholtz solutions using propagative plane waves suffers from severe instability in high-resolution Trefftz spaces, despite the existence of good theoretical error estimates. This instability arises from the redundancy of the approximation set, leading to an ill-conditioned linear system. As a result, regularization techniques become necessary. However, their effectiveness relies on the existence of accurate approximations with bounded coefficients in the discrete expansion – a property that propagative plane waves alone generally lack. To overcome this issue, we propose enriching the approximation set with evanescent plane waves, which are characterized by complex-valued direction vectors. These functions retain the algebraic structure of propagative plane waves, enabling closed-form integration on flat submanifolds – a key advantage in plane wave-based Trefftz methods. By incorporating evanescent plane waves, we achieve stability through approximations with bounded coefficients, thereby allowing for accurate numerical computations of Helmholtz solutions in finite precision arithmetic. Building on these results, we developed a Trefftz scheme based on evanescent plane waves with a conforming formulation, unlike standard discontinuous Trefftz methods. In this setting, the mesh is defined by the intersection of the domain with a Cartesian grid, and the basis functions are continuous, compactly supported, and can be expressed as simple linear combinations of evanescent plane waves within each element. This ensures they remain local solutions to the Helmholtz equation and allows for the exact assembly of the system matrix for polytopal domains. Numerical experiments confirm that the method achieves spectral accuracy while maintaining stability even in high-resolution spaces.