I think the easiest way would just be to try to expand whatever idea you have. If you want to Ne off of "hat", you need to branch outwards from the hat. You can branch into hat, hatmakers, history, mercury, lead, poisoning, arsenic, bottles, chemistry, bubbles, pasta, noodles, China, monsoon, umbrella, raincoat, hat, etc. Or you can branch into hat, hatband, ribbon, decorating, parties, weddings, white, sand, beach, ocean, nature, moon, stars, space, Neil Armstrong, Lance Armstrong, biking, helmet, hat, etc. The key is to keep moving, keep letting yourself jump immediately into the next idea without worrying about why or what it means or whether it's useful. What you eventually find is that you end up creating linking circles (like how I returned to hat both times), which gives you a greater concept-space around the word hat, and you can then more quickly relate hat to anything else in the world - and that begins to make it easy to see complex interrelationships that others don't necessarily track because you are so used to rapidly following strings of association and so habitually focused on reconnecting one thing to the next.
Sidenote, it's completely the opposite of Ni because the point is to do the opposite of distilling - you want to encompass as much as possible. Ni is looking inward at the hat and asking, what is the telos of this?, while Si is looking at the hat asking, what is the essence of this?, and Se is watching the hat in action and asking, what is this doing?. Ne is taking the hat, standing on it, and looking everywhere else.
