Gallagher, one of F1's longest-serving marketers, will leave Jordan at the end of the week after eight years with the team. He previously acted as a consultant to the team and worked at now-defunct F1 outfit Pacific Grand Prix.
Marketing responsibilities at Jordan will be shared between director of business affairs Ian Philips and Mike Hall-Taylor, chief marketing officer for Jordan Brand, the team's licensing and merchandising division.
Gallagher, regarded as having built one of the most solid set of sponsor relationships in F1, is joining the management team of Essentially Sport, a consultancy based in Gibraltar, but with offices in London. Other than Button, it counts former World Rally champion Colin McRae among its clients.
Taking on the role of director of marketing and communications, Gallagher will also become acting general manager of the firm's motorsport business.
His brief is to broaden Essentially Sport's remit across a broader portfolio of sports as well as diversify into other revenue streams to build its positioning as a one-stop shop for clients.
The company has just acquired the rights to F1 News, a publication it intends to relaunch under its Essentially Communications umbrella.