Although, I have no specific evidence relating to how the 2nd (Woodward's) was armed in the war's first year but I can say that most cavalry units in the Gen. A.S. Johnston's Army seemed to carry a brace of Colt Revolvers and a Double Barreled Shotgun. The Tensas Cavalry Company of Wirt Adams cavalry (1st Miss.) while in Ky were armed just so as was Terry's Texas Rangers (8th Texas.
There was very little uniformity however since most cavalrymen came into the service with their own personal weapons. One formal exception that I know of is Col. Forrest command at Hopkinsville in 1861. Forrest actually traveled up to Louisville and bought a significant number of Colt Revolvers (pistols) for his regiment.