I always ride with the halter under the bridle, lead rope attached to the halter and looped up to and slip tied to the saddle horn. I like to be able to controll Ming with a lead rope, rather than the reins. As it's been said, in case of an emergency, it's nice to have the ability to lead/tie/ or use for another horse, if necessary. It doesn't add that much weight, considering how nice it is to have.
When we take a break, I like to be able to pull off the bridle and let her graze - it she's behaved up to that point.
Looking at the trail through the ears of my horse...wonderful!