A removable case may receive an electronic device. A male connector in the case may mate with a female connector in the device. A battery in the case may supply power to the device through the male connector. The electronic device may have an antenna. The case may have a supplemental antenna that compensates for variations in performance in the antenna when the device is received within the case. The supplemental antenna may be a parasitic antenna resonating element that is formed from metal traces on a flexible printed circuit. The flexible printed circuit, a metal trim structure, and a plastic support structure may form portions of a connector support structure in the case.