There are several synonyms for the phrase "hand forward," including "extend," "proffer," "present," "offer," and "stretch out." These terms all refer to the action of holding out one's hand to offer something to someone, whether it's an object or a gesture of greeting. "Extend" suggests reaching out beyond one's usual reach, while "proffer" implies more formality or courtesy. "Present" is often linked with formal occasions, like giving a gift or award. "Offer" can be more casual or spontaneous. "Stretch out" is a less formal way of saying that one is holding out their hand. All these synonyms share the notion of reaching out to connect with others.