Dear Mr. Omansky,
Thank you for posting your request on History Hub!
We searched the National Archives Catalog and located a series titled Unit Histories, 1943 - 1967 in the Records of U.S. Army Operational, Tactical, and Support Organizations (World War II and Thereafter) (Record Group 338) but were unable to locate records of the 1588th Labor Supervision Company. We also located the World War II Operations Reports, 1940-1948 in the Records of the Adjutant General's Office, 1917-1985 (Record Group 407) that may include records of the 1588th Labor Supervision Company. For access to these records, please contact the National Archives at College Park - Textual Reference (RDT2) via email at email@example.com.
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We hope this is helpful. Best of luck with your family research!