In this work, we present the design and the experimental validation of a novel Bessel-beam launcher suitable for millimeter-wave applications. A zeroth-order Bessel beam is generated for the vertical component of the electric field. This is achieved by exciting a higher-order TM leaky mode in a grounded dielectric slab loaded with a capacitive impedance sheet. The distinctive feature of the proposed device is the use of a higher-order leaky mode: this allows scaling of the device to higher frequencies than would be possible using previous designs based on the fundamental leaky mode. As a result, thinning of the dielectric layer is avoided and ohmic losses are reduced. Simple design rules are provided for these types of devices. Measurements of a final prototype together with full-wave simulations validate the Bessel-beam generation in the desired mm-wave range.