Here's what I could find out-
Day's Cavalry was aka 12th Battalion, Tennessee Cav.- it's rosters shows the following:
Ellis, R. J.- Pvt. Co."E"
Ellis, James K.P.- Pvt. Co. "A"
Found no W.T. or varient name listed

For a year the 12th was combined with the 16th Battalion to form what was called "Rucker's Legion". The rolls of the 16th have 4 Ellis's including Pvt. William Ellis of CO. "F".

Sorry couldn't be of more help but it may give you some leads.