There are many support agencies dedicated to the office of president of the United States. The Secret Service use the coat-of-arms of the president on its uniformed officers, the White House Food Service workers have its own patch featuring the coat-of-arms of the president as well as the military personnel assigned to Air Force One, the helicopter Marine One and even the Washington, DC District Police use the coat-of-arms on its motorcycle presidential escort division.
Naturally, there are also many patches that were produced commercially that are not official. A few are designated here as well.
Many of these distinctive patches are shown here, but I’m sure there are others. Please drop a photo and description here, even if it is used unofficially.