Pony, the only real expertise I have is the period
circa 1858-1872, a bit before and after the War of the Rebellion. The normal, practical, expected, common garment worn under the fatigue blouse for that period is the issue wool shirt in gray, natural, or blue. Historical substitutes are home-made shirts in period materials and patterns, sutler- or store-bought shirts (ditto), something usable you "happened to find on your person while 'passing by' a civilian homestead" (this works especially well for those reenacting Sherman's march through Georgia
), or anything that your pay and/or circumstances might reasonably provide you.
Keep in mind, what you see in photographs of the period might well be a photographer's prop (paper collar, necktie, etc.). They were quite rare in the field, and I've long thought that vests are not really appropriate for enlisted men in the field as well. However, if you are a senior NCO who works at Division, Army or Corps HQ, in daily contact with senior officers, then I think making the effort to look "spiffy" would be perfectly appropriate. Just don't turn out looking prettier than the Commanding General - that's not a good career move.