According to FOLD3:
1 - The captain of Company E, 20th Texas Infantry was James H. McCardell.
2 - There is no veterans listed under the 20th Texas Infantry by the name of Cleveland.
3 - In all Texas units, there are 39 Cleveland's listed, but none with the first initial "E."
In the summer of 1861, many of the units organized were militia, home guard, or Texas State Troops. Later they were consolidated into CSA regiments. I have found several of these companies were organized by older men who did not go into CSA service or were not elected as officers when reorganized.
I checked Ancestry because they have cards on file from the state archives which includes the state units. No E. Cleveland found. I did find an E. Cleveland who enlisted in the 2nd Texas Cavalry (USA) and was promoted to Lieutenant. Rosters for the local units are few and far between. It makes them difficult to track.
The only other suggestion I could make would be to find a book(s) which list CW Veterans of Austin County. These type books are compiled by Ericson Books, county heritage books, genealogical societies, and sometimes local historians. They often provide information regarding organization of companies specifically from their county and the regiments they eventually became part of. The Clayton Genealogical Library in Houston has a ton of these type books and they are organized by county which makes them easy to locate. The problem is the library is closed at this time. The Heritage Museum and Research Center at Hill College in Hillsboro has an excellent collection CW information. It use to be called the Confederate Research Center until they decided to become politically correct. They are not open everyday.