On the Tishomingo County MSGenWeb there is a transcription of the "Enumeration of Confederate Soldiers and Widows" for the year 1911. On this enumeration there is listed J. H. Good with the following information:
Name: J. H. Good
State of Enlistment: Mississippi
Year of Enlistment: 1862
This transcription can be found at the following link:
Also, have you searched for a pension application file? There is a possibility that your ancestor applied for a Confederate Soldier's Pension. You didn't mention having searched for one.