I show that you are right in his regiment,he was in the first, You have the Day right July 4th but the year of the capture was 1863 and they were paroled July 7th 1863
I have a photo copy of the "Roll Of Prisoners Of War Captured" #43, where the POW's were signed in by Ink pen According to the 29 names on the list he was captured, Usually it flagged if there is POW, My GG Grandfathers served with him there, which is why i have the list.
If you like maybe i could email it to you,
Hope this may have helped some.
His Information is below, He did make it to Corporal
U.S. Civil War Soldiers, 1861-1865
about Thomas C. Williams
Name: Thomas C. Williams
Regiment State/Origin: Mississippi
Regiment Name: 1 Mississippi L. Art'y.
Regiment Name Expanded: 1st Regiment, Mississippi Light Artillery
Rank In: Private
Rank In Expanded: Private
Rank Out: Corporal
Rank Out Expanded: Corporal
Film Number: M232 roll 44