Veitch, who only joined FA in June 2024, will finish later this year after the game's governing body declared a $8.55-million loss in its annual report.
While on face value Veitch's exit may appear insignificant, it is the latest in a string of departures of powerful and experienced FA staff over the last year.
The highest-profile of which was the resignation in May of chief executive James Johnson.
Johnson has yet to be replaced with chair Anter Isaac promising FA would undertake a "global search" to find a replacement.
Former Matildas star Heather Garriock has been installed as an interim chief executive, but no candidates have yet to emerge to take the role on a permanent basis.
Whoever does become Johnson's long-term replacement will have a lot of issues to fix, as well as dealing with a mass exodus of staff from FA in the last 12 months.
Among them were marketing and communications boss Peter Filopoulos, who was moved on in September and suffered the humiliation of having his surname misspelt in an all-staff email sent by Garriock that announced his departure.
Filopoulos's media team has almost entirely left FA, including the media managers of both the Socceroos and the Matildas.
Chief of staff Alex Davani and legal boss Mel Mallam have also walked out of the game's Moore Park headquarters, while there has also been an exodus of coaches and technical staff.
Chief football officer Ernie Merrick was shown the door along with women's technical director Rae Dower.
Joeys (men's u17s) coach Brad Maloney resigned after returning from a tournament in China in June.
Ex-Young Matildas (women's u20s) head coach Leah Blayney has taken a role as a No. 2 with the Japanese women's senior national team, while former Matildas assistant coach Mel Andreatta became head coach of Scotland earlier this year.
Blayney has been replaced by Alex Epakis and Andreatta by Emily Husband, both of whom worked under Garriock when she was head coach of Sydney University in 2017.
Veitch's exit adds to the revolving door of departures and comes at a time when FA has lost a record amount of money over the last year.
Garriock said at FA's annual general meeting in May the sport's governing body would have to undergo a "right sizing".
That may have been made easier by swathes of resignations of executives and coaching staff, but FA is still yet to resolve issues which could have ramifications for the governing body's bottom line.
Most notably is the case taken to the Federal Court by A-League club Macarthur, who are seeking compensation of close to $10m.
As reported by AAP in April, Macarthur launched legal proceedings against FA, alleging the sport's bosses misled club owners when they applied for entry into the A-League.
