The US Army standard issue shirt was made of scratchy "domet" flannel, in an unbleached white (repros. avail from South Union Mills, Wambaugh & White, among others etc. etc.)
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They were rather disliked for being so coarse.
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Having once employed a domet flannel shirt, I used to wear a cotton shirt under it for mere comforts' sake. There are alternatives. County Cloth (Chas.R. Childs) sells the issue shirt pattern. Making one out of 100 percent wool flannel might not be 100 percent authentic, but it would probably be passably tolerable.
Other options...
The Army also purchased large numbers of contractor made flannel shirts in gray, blue, etc. Probably more comfortable.
Based on an original residing in a private collection in southern Ohio, this shirt represents just one of the three different styles of shirts issued to soldiers throughout the Civil War. Wool flannel shirts were manufactured by contractors and some arsenals throughout the war, and quartermaster...
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The Army also purchased large numbers of woolen knit undershirts...
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South Union Mills has some of this type made up from time to time.
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Where cotton or linen shirts were worn by the US Soldiers they were procured privately from sutlers, stores, home, or aid society distributions, etc.
Dr. Weaver reported that in exhuming and identifying Union dead at Gettysburg the undershirts were frequently the indicator where the bodies weren't wearing coats or jackets...
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The wearing of flannel shirts was common in the period. From 1853, recommending wear next the skin...
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Here's a check flannel shirt worn by Union Soldier Edgar Yergason...
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General Barlow customarily wore check flannel shirt too... here evidently with a paper collar (he WAS a general!)...
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A Union veteran, Musician Bardeen much later observed that he generally bought his underclothing outside of the quartermaster system.
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Bardeen mentions an occasion when some well meaning folks back home sent some "boiled" white dress shirts with fancy bosoms to their camp. The men had a laugh and ripped them up to use as cleaning rags.