The phrase "hold out one's hand" can be replaced with a plethora of synonyms based on the context of the sentence. Some of these synonyms include offering, extending, reaching out, proffering, presenting, and stretching forth. These terms imply the same gesture of reaching or stretching out one's hand to offer something or greet someone. The choice of the synonym depends on the tone and formality of the situation or relationship between the people involved. Whichever synonym is used, the message conveyed is the same, a friendly gesture to initiate conversation, or an offer of help or assistance.
