Sorry for the delayed response!
I have no knowledge of Jan Hagewolt. BUT...
Bergen's other book "REGISTER In Alphabetical Order, Early Settlers of Kings County, LONG ISLAND, N. Y" gave me a clue about the Lefferts of NY. In it he refers to Pieter and Leffert being assigned guardians on their father's death in 1662 which shows the must have been the younger two children on the Bonte Koe.
I then found a marriage record for Pieter in Bergen NJ:
1683 18 Nov; Pieter Pieterse Haeswout, ym, fr Dyckhuysen in Overysel; Dirckje Egberts, yd, fr Midwout on L.I. (c.)
On the ship's manifest the family is listed as from Drenthe. The geography of Drenthe and Overijssel is a bit of a mixture in this period, it's not unreasonable to state that they were from both. Now in modern Drenthe there is a village named Dijkhuizen... which is just over a mile from the tiny hamlet of Haakswold!
Haakswold is also known as Haagswold and "Haagwout" is one of the spelling variations used by the family in NY.
Personally i think this is strong enough evidence for the families' place of origin.