When tackin gup a horse, one of the most critical points is removing the halter to put on the bridle. because of the inherent nature of needing the head free for bridling, this is the most precarious moment of tacking up. Without a halter or bridle fully on, the rider has little to no control over the horse's head, and thus body, if the horse chooses to move.
To minimize some of the danger during these few moments, the best method is to unbuckle the halter while it is still tied, slip it off the horse's nose, and buckle it around the horse's neck. A panicing horse can easily slip out of this sort of confinement, but it is sufficient for the few minutes required for bridling.
The same method can be used when untacking your horse. Before removing the bridle, buckle the horse's halter around your horses neck, then remove the bridle, then slip the halter into place around the horse's nose.
riding with the halter under the bridle is a common practice of many trail riders but may not be the safest method. Halters are bulky and designed to move around on the horse's face. The bridle, however, is a much more specifically designed piece of equipment that should not be hindered by a bulky contrapsion slipping around underneath it. Additionally, the halter under or over the bridle may catch on the bridle and may inadvertantly send signals to the horse through the bit, or may create resistance blocking the reins from working correctly with the bit.