Charles II granted Sir George Carteret, Vice-Admiral of Jersey, land to settle in the Americas during his time in Jersey during the Civil War. The land granted was Smith's Island in Chesapeake Bay and was totally unsuited for settlement but Sir George dispatched settlers in the Gunderbark, when she was captured by a parliamentary frigate the Smith's Island project fell into abeyance.
After the restoration Sir George was rewarded with an interest in the new Carolina colony as were many of his relatives but there is no evidence of a flow of Jerseymen to Carolina.
After the Dutch were ousted from New Amsterdam their former area was spilt at the Hudson. The area from the Hudson to the Delaware was granted in part to Sir George and was named New Jersey, with, in 1665, Sir George's cousin Philip de Carteret arriving as governor. Philip named his capital Elizabethtown in honour of Sir George's wife.
Philip had brought with him a small party of settlers who seem to have been a mix of Jerseymen, English and Huguenots, but the majority of settlers moved in from New England at the behest of the governor of newly named New York who did not recognise the border at the Hudson. Tensions between the New Englanders and Jerseymen meant that the early years of the new colony were less than harmonious.
reference: p.270 "A People of the Sea" ed. A.G.Jamieson, Methuen 1986