Near the beginning of "The Inner Light", after Picard wakes up in Kataan, he goes outside and meets Batai in the square. After some questions he decides to take a walk outside the city, and you see him climbing a mountain bare-handed. The scene then cuts to a shot from behind him where he is looking out over the city; in his left hand is the hat that he doesn't end up with until later in the show when the sun's heat becomes increasingly unbearable. He didn't bring it with him.

In the very next shot when the camera is now facing Picard, he's holding nothing, doesn't pick up anything. The hat was never there which likely means they originally had him make the climb later on but spliced it in the wrong place.