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There are a few books available online that might help: Regimental History of the 316th Regiment of Infantry is one. The book, 79th Division, Summary of Operations in the World War should also help. It contains information about the order of battle and operations of the 316th.
Dear Mr. Schweiss,
Thank you for posting your request to History Hub!
We searched the National Archives Catalog and located the Records of Divisions, 1918-1942 in the Records of the American Expeditionary Forces (World War I) (Record Group 120) that includes Historical Decimal Files for 1918-1919 of 316th Infantry Regiment; and the Records of the 1st Through 338th and the 559th Infantry Regiment, 1/1/1916-12/31/1921 In the Records of U.S. Regular Army Mobile Units (Record Group 391). Some records are digitized and available in the Catalog but not those for the 316th Infantry Regiment. For access to and/or copies of records pertaining to the 316th, please contact the National Archives at College Park - Textual Reference (RDT2) via email at firstname.lastname@example.org.
We also located the series titled Draft Registration Cards, 1917-1918 in the Records of the Selective Service System (World War I) (Record Group 163) that are in the custody of the National Archives at Atlanta (RE-AT). For access to and/or copies of these records, please contact RE-AT via email at email@example.com.
On Ancestry, we located your grandfather Mauro Tummolo’s World War I Draft Registration Card. There may be a fee for using Ancestry. Instead, you may view these records online at one NARA's facilities for free via a NARA PC. For the nearest NARA location, please consult our web page at https://www.archives.gov/locations/. Also, some of the images from Ancestry are available for free on Family Search at https://www.familysearch.org/en/.
We hope this is helpful. Best of luck with your family research!