First off, you should not be focusing on the horse's literal "headset". What you are asking for is for him to collect his entire body, which starts at his hind end as he reaches under himself more in his strides. This leads to the back being more "rounded" and the horse moving in a frame. Once the horse is balanced and moving correctly, the head and neck will naturally set themselves.
Many riders, from beginners to expert dressage trainers, make the mistake of forcing the head and neck into a false bend. This actually hollows out the back and is the opposite of what you are looking for. Collection does not come from hard, oppressive hands or working the bit, but from teaching your horse to balance and building up muscle tone.
Maintaining light contact with your horse's mouth does help your horse to remain "on the bit". However, collection does not come from the reins alone! You need to have a light, flowing seat that encourages the horse's hind end to tuck in.
Of course, practice and exercise is the most important step in building that headset. As your green horse becomes more accustomed to riding and builds up muscle tone, he should continue to naturally reach down and lower/round his head.
Do not look for a "quick-fix" when it comes to collection! Just like with human athletes, horses need practice and steady work to develop their skills!
Feel free to contact me if you would like any more advice or if you have any questions!