Hammon will assume the role vacated by James Borrego, who was hired to be the Charlotte Hornets' new head coach May 10.
As ESPN's Adrian Wojnarowski put it, Hammon was promoted to the "front of the bench."
The 41-year-old Hammon has been part of San Antonio's coaching staff since head coach Gregg Popovich hired her in 2014. She was the first full-time female coach to collect a salary in NBA history.
Hammon notably led the Spurs to a title in the 2015 Las Vegas Summer League.
Hammon appears to be on the radar as a future NBA head coach, and her promotion led Chris Mannix of Yahoo Sports to wonder if she could eventually be Popovich's replacement in San Antonio.
Two weeks ago, Wojnarowski reported that Popovich isn't expected to coach in the league beyond 2020, the year in which he will lead the United States men's basketball team at the Summer Olympics in Tokyo.
He added that Popovich could spend 2019-20 preparing for that, meaning next season could be his final campaign on the Spurs' sidelines.
San Antonio has made 21 consecutive playoff appearances and won five championships under Popovich, and he boasts a career coaching record of 1,197-541.
If the Spurs do look for an in-house replacement, Ettore Messina is also a possibility, as he was the acting head coach during the 2018 playoffs, when Popovich took a leave of absence following the death of his wife.