In order to fully leverage idle computational resources in a fog computing scenario, a suitable discovery mechanism is needed. Within this context, F2C-Aware has been proposed in a previous work as a discovery solution where a device can detect nearby fog resources by scanning for custom 802.11 beacons being broadcast by those resources. Since mobility is an inherent characteristic in fog computing, keeping the Wi-Fi-based discovery service always on during mobility will lead to unnecessary scans being performed in non-covered areas, causing energy consumption penalties. That is why, in this paper, we propose a Beacon-assisted Direction-aware Scanning Scheme (BDSS), where we use the discovery beacon to convey the distance remaining until a next fog will be reached in each of the four cardinal directions along with the channel used for beacon advertisements. Therefore, the scan will be disabled until the next fog is reached. Emulation results show that by using this scheme, considerable energy savings could be made while maintaining a high discovery rate.