How do you increase the mechanical advantage of a pulley?

You need to add movable pulleys

The mechanical advantage of a pulley system is equal to the number of strands attached to the movable pulleys in the system. Consider the picture below.

As the string is lifted up, the pulley moves up, so this a "movable" pulley. Notice there are two strands of rope coming from the movable pulley. This means this system has a mechanical advantage of 2, so the force is doubled. There is an equal amount of force coming from both sides of the roped equal to the applied force.

By increasing the number of movable pulleys and strands of rope connecting the pulleys, we can increase the mechanical advantage. See the image below:

Notice in the image above there are two movable pulleys that are connected to 5 strands of string. Each of these pieces applies the same force coming from the hand. The mechanical advantage of this system is 5 (equal to the number of strands connected to the movable pulley) so it multiplies the applied force 5 times.

In the real world, carrying around tons of pulleys is not convenient. A block and tackle system is more useful. A block and tackle system are two compound pulley that do the same thing as multiple pulleys.

Notice how this crane hook is itself a movable compound pulley attached to 8 strands. The mechanical advantage of this block and tackle system is 8.