The excitation of phase resonances in dual-period wire structures under evanescent incidence is investigated and numerically reported for the first time. The diffraction problem of a compound structure comprising subwavelength perfectly conducting wires and slits near a dielectric interface is solved by the modal method, which has been extended to consider evanescent incidence via illumination under the total internal reflection condition. We demonstrate evanescent-to-propagating conversion in the transmitted intensity via resonances of different types, such as Fabry-Perot and Rayleigh anomalies. The results exhibit points in common with those obtained for simple structures (single wire/slit per period), but, on the other hand, there are striking differences, mainly arising from the excitation of phase resonances in the structure. These features allow an angular and spectral control of the transmitted response, which opens up new degrees of freedom for the design of input/output couplers and other devices.