We ride with the halter under the headstall too. It's just easier all the way around. No worries if you stop for a while, No worries if your tack fails, No worries about being worn out and slipping the bridal off and having the horse take off because it sees the green grass over yonder etc, etc.
The lead rope can also be used for a spare set of reins for a buddy that wasn't as prepared as you. I always try to be prepared...then you are in a better position to lend a hand too.
I normally trail ride with a rope halter underneath my bridle- you just never know when something is going to happen either in your group or someone else. We have come to the rescue a number of times and It always helps to be prepared.
I trail ride. Long rides, So we frequently stop for lunch or to just get off and stretch for minute. Somestime I tie the horse up and fish the rivers for bit. I leave a halter on under the headstall so I can take the bit out of their mouth, but still keep control of the horse. I use a rope halter on all my horses. I unclip and coil the lead rope up and tie it o my horn
I also carry a pair of Utah Hobbles on my saddles. For lunch we often will hobble the horses and let them graze while we eat lunch.
As an avid trail rider and endurance rider I use a biothane halter bridle. It is the best investment I have made in my tack inventory! The bit clips on/off without taking it off transforming it into a halter in seconds. The reins also have clips and can double as a lead rope in seconds.
PLUS, the biothane can be cleaned with soap and water, no leather to be cleaned or worried about in the rain/snow.
I only keep it on under my bridle if I think I'm going to need it a lot. Otherwise I wrap it up and tie it to my saddle, but if I don't think I'll need my halter, like on a short ride I just leave it at home. I use a bosal on both of my 2yr olds and the older horses just graze with thier bits in if we stop for a few minutes. Oh and I use rope halters and nylon but I prefer rope to nylon because they don't break and fit better under a bridle and are easier to take with you.