Boone did not identify the staff member, saying he was part of the team's support staff. Another staff member was not available because of contact tracing.
That brings the total number of cases among the Yankees traveling party to nine since Monday, including shortstop Gleyber Torres, who was placed on the COVID-19 injured list. Other identified cases were pitching coach Matt Blake, third-base coach Phil Nevin and first-base coach Reggie Willits, along with five staff members.
"We're just doing the best we can with it," Boone said. "Fortunately, he's another one that feels good. So we'll just continue to try and be vigilant and handle it as best we can."
The Yankees will have another staff member travel with the team to Texas. The latest staff member who tested positive is being quarantined in Baltimore.
On Thursday, the Yankees announced that Torres tested positive for the coronavirus despite being fully vaccinated and previously having COVID-19. He is quarantining in Tampa, Florida.
"I'm thankful that we have so many good people that not only have taken on different, more load or more responsibility, but also taking on some responsibilities that aren't necessarily in their expertise," Boone said. "We've had to share the load a lot. We've had people that have had to come here and plug right in, and really proud of everyone and how they've handled it."
In addition, outfielder Aaron Hicks was placed on the 10-day injured list Sunday, retroactive to Thursday, after he tore a tendon sheath in his left wrist. The club tried to treat the injury with medication. It's uncertain whether Hicks will need surgery.
"When he tried to swing from both sides of the plate, it wasn't what he was looking for or what we were looking for," Boone said.
In a corresponding move, the Yankees promoted outfielder Ryan LaMarre from Triple-A Scranton/Wilkes-Barre. He is starting Sunday's game in center field.
The Associated Press contributed to this report.