Many games on Civil war battles have game-turns corresponding to clock times. But commanders did not synchronize their watches as we see in later wars. So having battlefield events correspond to the clock times gives an incorrect impression of the ability to coordinate troop movements and attack times.
In a local setting, such as bivouac, camp, prison, or hospital the military day could easily be run by the clock, since these all required only one clock to control the events of the day. Hence the creation of public clock towers in small towns. In civilian life, just about every factory had its own clock tower that regulated working hours.
But in military situations that depended on multiple clocks over long distances, attempts at coordination only led to frustration. Every general's watch would be different, which would also differ from the tolling of bells in a nearby town clock. The sound of distant cannon fire trumped the tolling of clock towers in battlefield decision making.
It was mainly the railroads, which required that trains run on a schedule for collision avoidance, that caused the implementation of standard times and time zones. As long as a railroad ran on its own tracks and didn't share it with other companies, they could use their own company time in all their stations. At terminal stations that served multiple railroads, synchronization between companies would be essential.
Steamship departure clock times also important, but mainly for passenger convenience; the sailors were attuned to the lunar cycle of high and low tides.
For more information: Civil War Time: Temporality and Identity in America, 1861-