The autographed hybrid is being auctioned on ClubCharity.com through the Entertainment Industry Foundation, organizers said.
The proceeds from the auction are intended to support the Welcome Back Veterans initiative, which was created by Major League Baseball to assist veterans returning from wars and their families with deployment-related mental health concerns.
"Welcome Back Veterans is offering crucial support services to America's military men and women and their families," Hanks said in a statement. "When it comes to supporting our veterans and their families, I am proud to do whatever I can."
Hanks helped produce the military-themed TV miniseries "Band of Brothers" and starred in the Oscar-nominated World War II movie "Saving Private Ryan."