An electronic device may be provided with an antenna module. A phased antenna array of dielectric resonator antennas may be disposed within the antenna module. The dielectric resonator antennas may include dielectric columns excited by feed probes. A flexible printed circuit may include transmission lines coupled to the feed probes. The flexible printed circuit may have a first end coupled to the antenna module and extending towards peripheral conductive housing structures forming an additional antenna and a second end coupled to transceiver circuitry. Ground traces on the flexible printed circuit may be shorted to ground structures at the first and second ends to improve the antenna efficiency of the additional antenna. The flexible printed circuit may include an elongated slot with overlapping conductive structures and laterally surrounded by a fence of conductive vias to improve the flexibility of the flexible printed circuit while providing satisfactory antenna performance.