There are several synonyms for the phrase "make an additional demand". One such synonym is "raise a further request". This phrase conveys the idea of asking for something more, without explicitly stating that the previous demand was insufficient. Another synonym for this phrase could be "request supplementary requirements". This implies that more requirements are needed to complete a task or project. A more formal synonym could be "place an additional requisition" which suggests an official request has been made. Lastly, "demand additional services" is another synonym that could be used in a business context when more services are needed from a vendor or supplier.