Gear and clutch problems in the Osprey have killed more than a dozen troops in recent years, leading to groundings of the tilt-rotor aircraft.
The U.S. continues to strike Houthi militants, who have targeted Israel and American ships in the region.
President Donald Trump has said Caine wore the hat and expressed political support during a 2018 visit to Iraq.
Maj. Gen. Curtis Taylor, commanding general of the 1st Armored Division, said it would be a “long and difficult recovery operation.”
