I researched Jane Palmer Bevis on Ancestry.com and found some family trees that indicate that her mother was Elizabeth Adair, daughter of James and Hannah Adair of Laurens County. James died intestate in August 1818, leaving property to wife Hannah, seven children, and four children of Elizabeth Parlmore (sic) in the right of their mother. Hannah Adair died in 1826, leaving a will in which she stated: "It is my will after the following sums are paid to some of my grand children, to wit, to Patsey GAMBLE, the sum of fifty dollars to be paid to her when she comes of age. To Jinny BEAVERS one cow and calf, and to all the rest of my grand children whose parents are now dead the sum of one dollar." Jinny Beavers must be Jane (Palmer) Bevis. The other Palmer children, if still living, are not named. If Elizabeth Adair was the first wife of William Palmer, she would have to have died before 18 August 1803, when William Palmer married his second wife, Jane Gill. Elizabeth's children are not named in her parents' estates. I suppose it's possible that she could have been married twice and William might not be the father of all of them.
I recently took the Family Finder DNA test through Family Tree DNA. This test allows you to discover connections to descendants of all 16 of your great-great grandparents. Imagine my surprise when my closest match was to a 3rd great-granddaughter of Jane Palmer, born 9 January 1801 in Laurens County, South Carolina. The only possible way we could be related is if Jane was the daughter of my 3rd great-grandfather, William Palmer (1774-1840). Family lore has it that William was married twice, but it is said that he had two sons by his first wife, Joshua and Jerome. The 1800 census of Laurens County shows his family with 1 male 16-26 (William), 1 female 16-26 (his wife?), and 1 female under 10 (his daughter?). Depending on when the 1800 census was taken in Laurens County, the child might be Jane. William's mother's name was Jane.