Below the ))))) are some options to finding records.
Take care and hope this helps.
Contact the VA - Regional Office where he last lived.If they don't have any info, I think they pass it on to where he enlisted/discharged.Be sure to mention FOIA in your letter and in the address.FOIA is Freedom of Information Act.
Contact the County Clerk of Windsor County, VT to see if they have any information.
Contact the State Archives of Vermont to see if they have any records.
If he had a military funeral, contact the Sexton of the Cemetery and also the funeral home.I believe they have to have records of some sort that prove he was a veteran.
Now that you may have his Army Serial Number, contact the NPRC in St. Louis using Form 180.
They will give you the "Burned in Fire" standard answer.Keep after them.Since his name ends in C there is a very good chance they have his data. Resubmit using the additional paperwork they send you, give them some time.Then call or write them again in about 4-6 months and keep track of when you make contact each time and who you speak to.Sometimes they are able to recoup partial records.A retired LTC told me that the NPRC doesn't seem to interested in helping people trying to recoup WWII records.
Was he a member of any VFW chapter?If so, check with them.
Go out on eBay and search for 339th Inf Regiment and maybe something will pop up.I have had success with my search for history that way.Try your local or state library and maybe even the Library of Congress???
Here's a link giving info as to recovering records.
You can also write to the Department of Military Affairs in the State where your relative lived or served.
You can order copies of Veteran Records by writing to: National Archives Records Administration, General Reference Branch (NNRG-P), 7th & Pennsylvania Ave NW, Washington, D,C, 20408 or call (202) 501-5170 to ask for order forms and prices.
ARMY SERIAL NUMBER 31253876 NAME CROSS#MERVILLE#K######## RESIDENCE: STATEVERMONT RESIDENCE: COUNTYWINDSOR PLACE OF ENLISTMENTRUTLAND VERMONT DATE OF ENLISTMENT DAY15 DATE OF ENLISTMENT MONTH02 DATE OF ENLISTMENT YEAR43 GRADE: ALPHA DESIGNATION PVT#Private GRADE: CODEPrivate BRANCH: ALPHA DESIGNATION BI#Branch Immaterial - Warrant Officers, USA BRANCH: CODEBranch Immaterial - Warrant Officers, USA FIELD USE AS DESIRED TERM OF ENLISTMENTEnlistment for the duration of the War or other emergency, plus six months, subject to the discretion of the President or otherwise according to law LONGEVITY SOURCE OF ARMY PERSONNELCivil Life NATIVITYVERMONT YEAR OF BIRTH23 RACE AND CITIZENSHIPWhite, citizen EDUCATION4 years of high school CIVILIAN OCCUPATIONUndefined Code MARITAL STATUSSingle, without dependents COMPONENT OF THE ARMYSelectees (Enlisted Men) CARD NUMBER BOX NUMBER 0416 FILM REEL NUMBER 3.138