Given the closeness of Hamilton county, TN to McMinn county, TN I woud say the odds are that your Dye s may be one of the children of Henry Dye, whom Gilbert is the only one I have a source for. If one of Henry's children went with Gilbert, I would say they would appear on the 1840 census and maybe the 1850 census. I would also say they would be members of Pond Creek Baptist Church, McMinn county, Tn now First Baptist Church of Sweetwater, Monroe county, TN.
1. Willem Dey was a part of a group of migrants from the island of Barbados which means they were either less prosperous slave holders that sold out or most likely the children of indentured servants freed and given passage because blacks were cheaper, and mor heat/tropic disease resistant. He was probably born abt 1790 in Barbados. On the old stage road to Hackensack, NJ is a graffiti preserved by the State of New Jersey as a historical site that says "10 hour (by) stage anno (Domini) 1696. This would be the trip from New Amsterdam to Hackensack. A boy and his knife can do wonders on a rock outcropping! William and his five brothers were Torie supporters during the Revolution. They actually supplied food for them. The Dey s that stayed in NJ changed their names to Day to probably to help with the fact that their side lost. The ones that went to SC kept thier names because they were unknown
William He married Antje Jacobs.
2. Abrem Dey. Born abt 2/ Sep 1711 in Hackensack, Bergen County, New Jersey. Abrem died in Chester county SC, bef 27 Oct 1783; he was 72. He was christened in Dutch Reformed Church; Hackensack, Bergen County, New Jersey, on 9 Sep 1711
On 6 Jan 1737 when Abrem wae 26, he married Jannetje/Jane Ellis, daughter of Elias Ellis & Sarah Peers, in New Amsterdam, New York. Born abt 1715 in New Amsterdam, New York. she was baptized in New Amersterdam, New York on Nov 4, 1735. Jannetje/Jane died abt 1770; she was 55.
They had the following children:
i. William. Born abt 1739 in Hackensack, Bergen County, New Jersey. William was christened in Dutch Reformed Church; Hackensack, Bergen County, New Jersey, on 1 Apr 1739;
ii. Elias. Born abt 1741 in Hackensack, Bergen County, New Jersey. Elias was christened in Dutch Reformed Church; Hackensack, Bergen County, New Jersey, on 5 Jul 1741; he was <1. Elias married Maritje/Mary Cammega/Kammeyaer.
iii. Sarah. Born abt 1743 in Hackensack, Bergen County, New Jersey. Sarah was christened in Dutch Reformed Church; Hackensack, Bergen County, New Jersey, on 3 Jul 1743
iv. Annatje. Born abt 1745 in Bergen County, New Jersey. Annatje was christened in Schraalenburg Church; Hackensack, Bergen County, New Jersey, on 3 Jun 1745
3. v. Johannes (John). Born abt 14 May 1748 in Hackensack, Bergen County, New Jersey. Johannes was christened in Dutch Reformed Church; Hackensack, Bergen County, New Jersey, on 22 May 1748
Johannes married Agnes McKeowan. Born abt 1754 probably South Carolina. Agnes died in South Carolina, abt 1859
vi. Jacob. Born abt 1750 in Bergen County, New Jersey. Jacob was christened in Schraalenburg Church; Hackensack, Bergen County, New Jersey, on 1 Jul 1750
vii. Richard. Born abt 1752 in South Carolina.
viii. Robert born abt 1754 in South Carolina.
4. Henry Dye born abt 1771 Chester county, SC died 14 Apr 1808 in Rutherford County, SC. Married Anna Washborne born 1771 probably Chester County, SC died 10 Aug 1853 Ozark county, MO, buried in Herndon Cemetery. She outlived her grandson Gilbert by six months and was living with him when he died. She lived about less then a month after moving to Ozark county MO.
5. Gilbert Dye born 5 May 1803 9 Rutherford, NC Died Feb 1853 Mcminn county, TN ( This is the same location that John and Mary Seek Fry, his son Henry Fry and Alpha Rutherford were living per 1840 census, 1850 census, and records of pond Creek Baptist church, now First Baptist Church, Sweetwater, TN) He maried Priscilla Harrill born 20 Jan 1806 probably Rutherford county, NC died 7 Nov 1876 in Almartha, Douglas county, MO biried in the Frye cemetery near Almartha, MO
Gilbert Dye's, who died in MicMinn county, TN, daughter Mary Ann Dye married Azariah (Zear) Olden Parnell who lived in Mcminn county, TN fpr about a year then moved to Kentuckey. Unknowingly then he moved to Ozark county within 30 miles fo the Fry s their daughter Nancy Pricilla Parnell married James Marion fry who's father Harvy Fry lived next to Gilbrty Dye in McMinn county, TN.
Type in "The dye society" in google search and ask your question.