With the commutation system still in force the Wilkinson Invincibles probably got their first uniforms or clothing from the local community. Contemporary local newspapers may give you an indication of the sorts organisations and individuals who provided this sort of support during this period. Interlibrary loan is a good way of accessing these. Later in 1862 they were amongst troops who wore Federal uniforms ransacked from stores during the Antietam Campaign. During the period Oct 1863- October 1864 the QM of Georgia issued their regiment 20 hats, 100 coats and 100 pants. There is some evidence that these the coats were frock coats. I suggest a scan of the regiment's Compiled Service Records - take especial note of the records of the regiments QM and for company commanders. If the microfilms of these aren't available at your local library, then inter-library loan is a great way of getting them. Happy hunting.
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