In addition to Walter Wilmoth in Tennessee you might also contact Dean Wilmoth in Nebraska at firstname.lastname@example.org
Dean is a "cousin" of my husband and a lot of info on the Wilmoth family that Walter does not yet have but we are in hopes of tying our families together somewhere back.
In the meantime, I will check in all of our Wilmoth info.
Have you tried the Social Security Death Index? Do you have a copy of the death certificate for your father? Do you know where he died and was buried? If you can get any of these, you will get a lead to get his birth certificate.
I live about 25 miles from downtown Washington DC and get into DC every once in a while to do genealogy research and if I can ever be of help, like once you get some info as a lead, don't hesitate to let me know.