@Scott,
Indeed. I was thinking about different scenarios where it would have been possible to do.
You are quite right about it being impractical if not impossible while on the march. As a reenactor of 25+ years I saw firsthand how difficult it would be to not only strip off all my gear and then put it back on again, but as you also point out – where do you carry your coat or jacket? The only thing I can think of is that they would be piled in the rear of the regiment with the intention of coming back for them later.
Knapsacks were frequently dropped in this manner, where a regiment would pile them and plan to come back to pick them up later on. Of course, that plan would be totally useless is the enemy happened to overrun the area where they had left them!
I was thinking more along the lines of it happening before a regiment broke camp in the morning. Or perhaps in the Western theatre where there are some references to Western federal troops going into battle in shirtsleeves.
I would agree that it would have to be a decision made beforehand, whether it be a battle or a march.
If memory serves, the Troinani painting depicting the 9th Mass. in shirstsleeves at Gaines Mill is based on letters and records that refer to it being done prior to battle.
As you quite correctly pointed out, there were no baggage wagons committed to carrying any of the soldiers' actual uniform items or gear, so if a soldier dropped his coat, or any piece of gear individually, it would be lost.
And of course there was the practical reality of the government charging you against your personal issue account for any uniform items lost. I remember reading that this became very problematic for federal soldiers who had dropped gear on the march to lighten their loads, or had in some way or another lost a cap, hat, or canteen in any number of numerous ways, and were charged for them. I remember reading that even soldiers who had lost hats or caps in the course of a panicked retreat, where like a tree branch might have knocked their headgear off. Apparently there were few if any exemptions where you could avoid being charged.
I do remember a long while ago seeing a company of reenactors who had slung their fatigue blouses over their waist belts and pushed them around to the rear, so that the blouse was draped over the belt.
I also recall seeing reenactors who had rolled their blouses and frock coats into rough cylinder shapes and had strapped them to the tops of their knapsacks.
Those are both possible ways that Civil War soldiers might have done it as well, but of course it's all pure speculation.