There are several synonyms for the phrase "hold out a hand." One common synonym is "extend a hand," which implies offering help or support. Another option is to "reach out" to someone, suggesting a desire for connection or communication. You could also "offer a hand" or "lend a hand," which denotes a willingness to assist.
In a more formal context, you might use the phrase "extend an invitation," which is a way to offer a hand in friendship or hospitality. Alternatively, you could "proffer a hand," which has a more formal tone and suggests a desire to make a formal introduction.
No matter which synonym you choose, the idea of holding out a hand is one of kindness, generosity, and support for others in need.