I read through your postings on some of the other forums and saw that you had more information than was listed here.
This is likely your father's Social security Application: From the Social Security Death Index Jessie Jones, born 8 Dec 1924, died Jul 1985.Last known residence: Cuney, Cherokee, Texas.SSN Issued Texas (Before 1951).SSN 461-30-8180
I would recommend that you order this from the Social Security Administration.There is a fee, but your father filled this application out on his own, and it should list the name of his father.
Since you have birth dates for Hattie Lorene and Edward, I would try to get their birth certificates from the county where they were born.If you know their death dates, you can try to order their death certificates or Social Security Applications too.
Texas birth records from 1926 - 1950 are available on usgenweb.This link will allow you to conduct searches for all of Texas: http://www.rootsweb.com/~usgenweb/ussearch.htmhttp://www.rootsweb.com/~usgenweb/ussearch.htm You may be able to find Edward listed through here, and then get the name for his father that was used on his birth certificate.
Fannie's death certificate should also list the name of her husband.
You could also look for the marriage certificate of Fannie in the county where the first child was born (I would search between 1920 and Mar 1922).
I tried searching in the 1930 census for this family but didn't have any luck searching on the children's names.They may have been "hiding" with some of my missing ancestors ;)