Directional antenna systems are gaining widespread adoption in wireless communication solutions, particularly using super-6 GHz bands in the electromagnetic spectrum. Hence, neighbor discovery and beam alignment in these directional wireless systems have attracted notable attention from researchers. However, fast neighbor discovery using directional wireless while maintaining covertness from eavesdroppers by minimizing the probability-of-intercept (POI) is an open research problem. We address this trade-off by proposing a sequential transceiver (or direction) selection protocol based on a tuning parameter (α) that guides the nodes to prioritize either goal by selecting the next operational transceiver subset for probing. We consider a 2-D multi-sector directional wireless system with electronic steering of transmission among sectors, assuming each sector is equipped with a transceiver and the sectors collectively cover the 2-D 360 • horizon around the node. We design a time-slotted neighbor discovery protocol that employs a probabilistic approach to select only one transceiver to use for the next time interval. By changing α, we study its capability to control the probing direction and impact on neighbor discovery speed and POI. Results show random selection offers fastest neighbor discovery but increases vulnerability to passive eavesdropping. Choosing from transceivers placed opposite to the active one accelerates discovery, while prioritizing the adjacent ones enhances covertness. During the search for the best α, we observe that exclusively prioritizing either rapid discovery or minimizing POI for a long stretch does not yield an optimal solution.