The problem with that "rebirth" is that in the "Year Without Summer"
(The Year Without a Summer, also known as the Poverty Year or Eighteen
hundred and froze to death, was 1816, in which severe summer climate
abnormalities destroyed crops in Northern Europe, the American
Northeast and eastern Canada. Historian John D. Post has called this
"the last great subsistence crisis in the Western world." - Wikipedia)
it was recorded that the "mechanics" those that built ships in
Setauket, had to wear their coats all summer. So a "rebirth" in the
1840s probably refers more to the number of shipyards, perhaps rather
than the industry which might have been there for a while. The 18th
century reference is in regards to an original settler who was thought
to have violated some principle and allowed to stay long enough to
build a boat to leave.

In the 17th century, John Scot, whom one historian has called the
"first president" in America, as he represented over 200 Long
Islanders, when he presented a petition to the re-established Dutch
government in New Amsterdam (New York) for religious freedom, and not
the sworn unquestioning allegiance to the Prince William of Orange
(later William of "William and Mary" of England) the petition which
was torn-up unread in public. He was arrested in Setauket and through
a deal with the Connecticut government imprisoned there. After a year,
his wife came to visit, apparently about to give birth, and as the
story goes, after snickers, allowed to see her husband John Scot.
Under her dress was a long length of rope, with which John Scot, a
native of eastern Long Island, escaped never heard from in history
again. Perhaps he is the origin of the usage "scot free"? Today it was
said to be the first use of O.K. in America for Old Kinderhook where
President Martin Van Buren had his summer White House, and when asked
where the President was "O.K." I was part of a crew that went on to
test for the "summer White House" there for the National Park Service
now restored, its Italianate bell-tower returned apparently to its
mauve and maroon colors.