See Sandra Riley,Homeward Bound: A History of the Bahama Islands to 1850 with a Definitive Study of Abaco in the American Loyalist Plantation Period.(1983, 2000)
On page 29, Riley writes that “Some time between the spring of 1646 and the fall of 1648, William Sayle set out with seventy prospective settlers, a group composed of religious Independents from Bermuda and some people from England.”...His ship was named the William.They reached Bermuda.The William wrecked.Provisions were lost.On page 30, “Unable to stand by and watch his people perish, Sayle provisioned the shallop as best he could and, taking eight men, set out to find help.After nine days their supplies were exhausted, but Sayle reached Virginia where religious sympathizers furnished him ‘with a barque and provisions for the relief of the settlers in the Bahamas.’”
Fortunately, Riley’s book has been copied by Google and is available for review on the internet.Just Google the book’s title. Unfortunately, important citations - such as footnote 25 - do not appear in the Google version.The Riley book doesn’t prove a connection between Captain Sayle and William, the immigrant, but the reference may justify further research along these lines.