An electronic device may be provided with a sensor module and an antenna having an antenna arm, ground structures, and a tuner. The tuner may be mounted to a printed circuit overlapping the sensor module. A spring may be mounted to the printed circuit and may couple the tuner to a conductive chassis of the sensor module. The sensor module may include optical sensors that gather sensor data through a display and may form ground paths from the tuner to the ground structures. Conductive interconnect structures such as springs may exert biasing forces in different directions to couple the ground paths to different layers of the ground structures. This may serve to couple the antenna to the ground structures as close as possible to the tuner, thereby maximizing antenna performance, despite the presence of the sensor module.