The phrase "go off at half-cock" means to act impulsively or prematurely without sufficient preparation or consideration. There are several synonyms that can be used instead of this expression, including "act rashly," "jump the gun," "be hasty," "act prematurely," or "be impetuous." These phrases convey the same meaning as "go off at half-cock" and can be used interchangeably depending on the context of the situation. It is important to choose the right synonym that fits the context, as this will ensure that the intended message is conveyed accurately.