[Arwen Evenstar the Fair]

Hey guys, I was just wondering, you know when Gandalf is on top of Orthanc and then he jumps on the King of eagels back, ok well, notice he doesn't have his hat anymore cuz it fell off when he was fighting with Saruman. But when you see the fellowship leaving on their quest, he has it back. How did he get it back?? Did he go back to Saruman's place, knock on the door and be like, "Oh yea one more thing I forgot my hat, can I have it back please??" I mean, come on.