BORN in 1562 (source Hannah French.ged) or between 1552-1581 (source Brøderbund WFT - a collection of gedcoms)
MARRIED 1587 (to Olmstead, source Hannah French.ged) or between 1593-1610 (to Anne Unknown, source Brøderbund WFT - a collection of gedcoms)
DIED 27 Jan 1613 in Essex, England (source Hannah French.ged) or 05 NOV 1639 in Massachusetts (source Brøderbund WFT - a collection of gedcoms)
Thomas had 12 children shown, with birth years between 1590 and 1605.
The French Family Association shows:
3.3 Thomas French, Jr., chr. ca. 1551, m. Anne or Agnes Olmstead in 1587 in Farnham, Thomas d. 1613 in Halstead. Agnes Olmstead b. 1566 in Halstead, d. 11 Feb 1623/24
This is an earlier birth year, and the same marriage and death years as some gedcoms.
For the parents of Thomas, the French Family Association shows:
2.6 Thomas French Sr., chr. ca. 1530 Arkesden or Wethersfield, m1. Elizabeth, m2. Bridget, Thomas d. 1599. Q/SR 4/7 on 6 March 1561 was accused of breaking into the close of William Waldegrave, esquire, as Great Wenden and cutting down and carrying away 2 cart-loads of wood.
Thomas m1. ca. 1557 Elizabeth and had at least 3 children (Mary, Elizabeth, Thomas).
Thomas m2. Bridget Burrough, born Hingham or Heigham, on 2 Dec 1597   when he was about 67 years old. She was the widow of Thomas Burrough of Wickhambrook, Suffolk, as his second wife ca. 1564 in Wickhambrook ; his first wife being Elizabeth Burrell. Bridget was b. ca. 1542 in Higham, Suffolk, England , the dau. of Thomas Higham and Phyllis Waldergrave . At their marriage, she was about 55 years old; therefore, no children were born to this couple. Their marriage ended 6 months latter when Bridget died. Therefore, in the year 1597, Bridget’s first husband died, she remarried, and then she died herself, all in the same year. Two years later Thomas French died. Thomas Higham d. ca. 14 Dec 1559 in Gazeley, Suffolk, England. Bridget d. 19 Jun 1597.
According to Douglas Richardson's Royal Ancestry, Vol 2, on pages 44 and 45, Bridget Higham was married (1st) to Thomas Burrough, Gent. Thomas Burrough died 19 Jun 1597, buried in Suffolk. Bridget married (2nd), in Suffolk, on 2 Dec 1597, Thomas French, Gent., who left a will proved 31 Oct 1599. Richardson also does not show any issue for Bridget Higham and Thomas French. Richardson also does not mention that this was a subsequent marriage for Thomas French. Richardson cites (1) Muskett, Suffolk Manorial Fams. 1 (1900), p 310-314 (2) NEHGR 108 (1954), p 172-178. Harvey Visitation of Suffolk 1561 2 (H.S.P. n.s. (3) (1984): 391-399.
Disputed Mother Conclusion
With the marriage of Bridget Higham to Thomas French in 1597, that would have been 35 or 46 years before this Thomas' birth based on 1562 or 1551.
Based on these sources and the timing of the birth/Christening of this Thomas and the birth years of his children, Thomas' mother should be Elizabeth (Unknown), not Bridget Higham
Christened circa 1551
Birth estimate: Bet 1552 AND 1581 
Birth about 1562, Wethersfield, Essex, England
Birth ABT 1570, Assington, Suffolk, England
Died 1613 in Halstead
Death: 27 JAN 1613, HALSTEAD, Essex, Eng
Death: 27 JAN 1613 
Death: 05 NOV 1639, Ipswich, Essex, Massachusetts
Married Anne or Agnes Olmstead in 1587 in Farnham
Richardson, Royal Ancestry (2013) Douglas Richardson, Royal Ancestry: A Study in Colonial and Medieval Families, 5 vols., ed. Kimball G. Everingham, (Salt Lake City, Utah: the author, 2013), volume II, pages 44, 45, BURROUGH 17, entry for BRIDGET HIGHAM (This source shows this Thomas French-266's father married Bridget well after Thomas' birth, supporting HIGHAM-55 should not be the mother of this Thomas)