How do you improve on the flexed arm hang?

You can improve by increasing time or increasing resistance or some combination of the two. Pull a fixed part of yourself to touch a bar (if possible) like your chin, neck, or sternum. Maintain contact for a fixed period of time. If counting to oneself is not available and you don't have a partner, try an egg timer and aim for a certain time.

You can also add a weight to various areas. In the case of a dumbbell, you might hold it for a period of time, and set it down and then hang with just your weight for a period of time.

Alternative, if you need assistance, hang with your bodyweight for time and then put your legs into stirrups or a hanging stretch band to take some weight and hold for additional time.

You can also train for this by holding a lat pulldown bar in the bottom position for time.