During and right after the Civil War, the Army consisted of 5 regiments of Cavalry, 10 infantry regiments of 10 companies each, 9 infantry regiments of 3 battalions with 8 companies each, 5 cavalry regiments and the Corps of Engineers. The corps and divisions of Civil war fame, or those organized for the Spanish War went away.
After the Civil War, the US Army expanded. This was the only peacetime expansion of the army in US history.
Under the Act of 28 July 1866 the army consisted of 10 regiments of cavalry, 5 regiments of artillery, 41 regiments of infantry and 5 regiments of of the Veteran Reserve Corps.
By the Act of 3 March 1869, the infantry was reduced to 25 regiments and the Veterans Reserve Corps went away.
By the Act of 8 March 1898, artillery was increased to seven regiments.
By the act of 9 February 1901, 15 cavalry regiments were authorized, infantry regiments were expanded to 39 and the artillery became a corps, losing its regimental stucture and consisted of 30 field batteries and 126 coast artillery companies.
In between each act, other legislation raised or lowered the number of men in companies/batteries/troops as well as the the other branches. They also added corps such as the Signal Corps.
The number of companies in a regiment varied from 10-12 depending on the period.
There were no tactical divisions or corps. There were administrative entities called divisions that had geographic boundaries but the divisions like the 1st Infantry Division didn't come on the scene till 1916. There were also corps, ie., the Corps of Engineers or the Signal Corps.
In fact, even today, the only permanent organizations in the army are its regiments and corps (again, like the Corps of Engineers)and Special Forces Groups. I doubt a US Army regiment has been disbanded since WW II. They maybe inactivated, but their colors are returned to the Department of the Army and they maybe reactivated. The 24th Infantry, organized in 1866 and deactivated in 1951, was reactivated in the 1990's.
A good treatment of the period you're interested in is in "Historical Dictionary and Biographical Dictionary of the US Army", by Francis B. Heitman (GPO, 1903 and many reprints), Vol. 2, pp. 598-623. Organization is laid out in excrutiating detail in this source.
James B. Ronan II
To Join the Company of Military Historians click here