Weirdly, there are actually a couple of records for Fincher contained within Earl's file.
There is a card showing that "Jno. P. Fincher" of Co. F received pay for two months' service (from April 30, 1863-June 30, 1863).
He received $22 for these two months of service, so he was at least in active service for the time period indicated.
There is also a reference to a document that might shed light on his service.
"Title of Record" is "L.R. #709, Dept. Ala. Miss & E La." and dated June 24, 1863.
(The "L.R." might be "letter record" or "letters received," though I am just speculating.)
The subject line reads "requests to be dischg."
The soldier is referenced as "Fincher, Pvt. Co. F, 19th La. Regt."
The document itself is not found in either Fincher's or Earl's file.
You can often get these documents by writing the National Archives and Records Administration in DC, though I suspect that it would be easier for you to request them through the hosts of this website via the link at the top of this page.
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Hope this helps! -- Jim Huffman, Co. K, Violett Guards, 6th LA Inf.