His family, of Irish origin, were the proprietors of the Hennessy cognac business, now part of LVMH.

Hennessy was re-elected in the election of 1928 and served as agriculture minister from 1928 to 1930, and then as French ambassador to Switzerland.

His elder brother, James Hennessy had been elected before him to Parliament as député and senator, but he chose to dedicate himself to the management of the family business.

On 10 July 1940, at the joint session of the French parliament which granted strong powers to Marshal Philippe Pétain, Jean Hennessy and his brother James were part of the 80 parliamentarians who opposed the measure.

He had been married to Marguerite de Mun, who bore him two sons, Patrick and Kilian, and a daughter Jacqueline.