Suppose I put the hat on my hat very loose, then I jump as high as possible, what would happen to my hat?
I think you probably mean put the hat on my head since the state of the second hat is not mentioned. If you are jumping, the hat is likely to be pulled off by the air resistance as you begin to fall and slowly fall back later. However, the result can be influenced by the weight and surface area of the hat, as well as how high you can jump. For example, if the hat is almost the same size as your head, it may not experience enough air resistance to slow it down obviously.
What does the weight have to do with it? How does the weight of the hat influence the hat falling off or not? Can you explain this further?
Sure. To fall off your head, the hat needs to go down slower enough, thus it can be left back enough from you when you land. The deceleration from the air resistance can be estimated as F/m, where F is the drag force and m is the mass of the hat. So if the hat is too heavy, it can't go slow enough to fall off.
For example, if you drop a piece of paper, it will drift and fall slowly. But if you drop a steel panel of the same size, it will fall much faster under gravity.