it's for this very reason that I think it's not good to teach a horse to come down off the bit into a "headset" by lowering the hands and fiddling the rein, then releasing when he's there, which sounds like what you have been doing. Now, your horse is used to that and does not know how to accept bit contact. When you lower your hands and fiddle him down, he is coming behind the bit, in a way. In that , he comes behind and down off the bit until you release, which says to thim "right here".
The way to be able to put and keep your hands up where they belong is to teach your horse to be able to accept contact with the bit. To do that, you have to only give a release when he is 'on' the bit, whether it's up or down.
For this, I would do excersizes riding him around and getting him to make contact with the bit, giving a small release , then taking up contact again and asking him to kind of "push" the bit out and forward, but not drop it or come behind it. And you never achieve this by lowering your hands or pulling him down. YOu might very softly milk the rein a tiny bit, just to give him the idea to focus on the bit, at first, but you try not to give up the correct elbow to bit line, even as the horse starts to lower his head.
Think of your thumbs as laser sights that come out of your straight wrist, and laser in to the bit. If you drop your hands, you will be lasering the ground, not the bit.