Essentially nonoscillatory (ENO) and weighted ENO (WENO) methods on equidistant Cartesian grids are widely used to solve partial differential equations with discontinuous solutions. The RBF-ENO method is highly flexible in terms of geometry, but its stencil selection algorithm is computational expensive. In this work, we combine the computationally efficient WENO method and the geometrically flexible RBF-ENO method in a hybrid high-resolution essentially nonoscillatory method to solve hyperbolic conservation laws. The scheme is based on overlapping patches with ghost cells, the RBF-ENO method for unstructured patches and a standard WENO method on structured patches. Furthermore, we introduce a positivity preserving limiter for non-polynomial reconstruction methods to stabilize the hybrid RBF-ENO method for problems with low density or pressure. We show its robustness and flexibility on benchmarks and complex test cases such as the scramjet inflow problem and a conical aerospike nozzle jet simulation.