I am a member of the Winn Genealogical & Historical Association and have access to some Winn Parish historical data. Gregg Davies of Winnfield may also be able to help.
I just looked in Booth's "Records" and found two men surnamed LAWLEY [Elijah LAWLEY and William LAWLEY] who were members of Company F, 31st Louisiana Infantry. These are the only two men surnamed LAWLEY listed in Louisiana Confederate service. Elijah enrolled at Vidalia [Concordia Parish], Louisiana on April 26, 1862 and William enrolled at Jackson, Mississippi on May 4, 1862. Elijah was captured and paroled at Vicksburg on July 4, 1863. He has no record of exchange or further service and no final parole. William's service record simply ends with "present for duty" on the January/February 1863 muster roll.
Dr. Bergeron's "Guide to Louisiana Confederate Military Units 1861-1865" (LSU Press, 1989) identifies Company F, 31st Louisiana Infantry as the Catahoula Avengers from Catahoula Parish. Vidalia is on the Mississippi River in Concordia Parish directly across from Natchez.
We have a listing of Union veterans from Winn Parish published in "Legacies & Legends of Winn Parish", a quarterly journal of the Winn Genealogical & Historical Association. The surname LAWLEY does not appear in these listings.
I also looked through published extracts of the 1860 and the 1870 Winn Parish Federal census without finding anyone with the surname LAWLEY residing in the parish either before or after the war.
Questions: (1) How old was Frank LAWLEY and (2) why do you think he was in Winn Parish? Age is an issue because this man may have been too old or too young to have served. Passage of the Confederate Conscription Act of April 1862 required enrolled Confederate service of white male residents ages 18 to 35 years. In September 1862, the upper age limit was raised to 45 years. In February 1864, the age limits were extended to cover men 17 to 50 years.
Those men of the 31st Louisiana who went home to Louisiana on parole after the fall of Vicksburg in the summer of 1863, and who reported into parole camp during the spring and summer of 1864 when ordered to do so, were exchanged and returned to duty in August 1864. The 31st Louisiana was re-organized and remained on duty in Louisiana for the balance of the war. Some new recruits may have enrolled during this final period of the war and not reported into Federal parole centers after May 26, 1865 for a final parole. This would explain an absence from the Confederate service records.
I recommend you search the 1860 Federal census records covering Catahoula and Concordia parishes based on the two LAWLEY men who served in the 31st Louisiana. The surname appears to be rare and these Confederate service records suggest a family cluster in this area of the state. The following Internet sites should put you in touch with researchers familiar with these two parishes.
http://www.rootsweb.com/~usgenweb/la/lafiles.htm
http://www.geocities.com/robertce/catahoula/index.html
http://www.rootsweb.com/~laconcor/