When looking for synonyms for the term "more hawk-eyed," several options come to mind, depending on the desired context and tone. For example, one could use "observant," "watchful," or "alert" to indicate a similar level of attentiveness and focus. Alternatively, "sharp-eyed," "keen-eyed," or "piercing" could suggest a more probing or piercing gaze. Other options might include "discerning," "perceptive," "insightful," or "discriminating," which could convey a sense of intelligence or understanding as well as vigilance. Ultimately, the choice of synonym will depend on the speaker's intended meaning and the level of intensity they wish to convey.