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.
Be strong and courageous, and do the work. Don't be afraid or discouraged by the size of the task, for the Lord God, my God, is with you. He will not fail you or forsake you.
1 Chronicles 28:20