Jones, who has been CFO since February 2016, takes the executive director position of chief operating officer on the Huntsworth Board.
Head of finance Ben Jackson replaces Jones as CFO.
Huntsworth CEO Paul Taaffe said: "Huntsworth has grown rapidly over the last two years and these appointments will allow us to scale even further.
"Neil has already proven to be a very commercial chief financial officer and as chief operating officer will focus on improving all operational aspects of the group and driving growth. In his new role he will have specific responsibility for the Communications Division.
"Ben joined the group three years ago as head of finance. In the past year, he has been taking an increasingly prominent role in the group’s senior financial decisions, banking relationships and control environment improvements, and we are pleased that he will now take on all aspects of the chief financial officer role. "
Huntsworth has seen an improved performance in its comms agencies recently.
The firm saw like-for-like revenue growth in its Communications arm for the first time in "a number of years" in the first half of 2019, with Grayling UK highlighted for its "standout" performance. Operating profit and margin in the division also grew although overall revenue was flat.
Last month, Grayling announced the promotion of Sarah Scholefield and Richard Jukes, chief executive and chairman respectively of the UK business, to lead the agency across Europe following the departure of Europe CEO Jan Simunek.