The reason that s.r. are first put onto a caveson is so that the horse will work into a connection/light press so that there is NO pain on the bars. And then when they are connected to the bit (appropriate height, not set low), th ehorse should NOT put its head down. That is a mis-application of s.r/draw reins/etc.
IF one puts only one side rein on (outside) I would NOT recommend stringing the line through the surcingle rein, since that can pull the horse around in a circle and get the handler in real trouble if they do not have good lungeing skills. Just go straight from the caveson (or inside bit ring...ideally a D-ring, full cheek or fulmer) to the handler's hand.