The phrase "pressing one's hand" can be replaced with a variety of synonyms that convey a similar action and intent. For instance, one could say "clasping someone's hand" to denote the act of holding onto it firmly. Alternatively, "squeezing someone's hand" could indicate affection, greeting, or even a signal of reassurance. Other synonyms for "pressing one's hand" might include "grasping," "clutching," or "taking hold of." Each of these phrases can convey different nuances, such as the degree of force applied or the emotional connotation behind the action. Depending on the context of the situation, different synonyms may be more appropriate to use.