The phrase "hold out one hand" has several synonyms, including "extend one's arm," "offer one's palm," "present one's hand," and "proffer one's digit." These terms are all used to describe the act of offering a hand to someone else, either in greeting or for assistance. "Extend one's arm" is a more general phrase that refers to the act of stretching out one's arm, while "offer one's palm" specifically refers to showing the flat part of one's hand. "Present one's hand" is another common phrase for offering a handshake, while "proffer one's digit" is a more formal and old-fashioned way of describing the same action.