This is a state that very, very few Putnams and Putmans ever set foot in. And, just in case you were interested in the first place, all the census records from 1790 until 1830 were stolen, burned or just buried in the Secacus Meadowlands when they built Giants Stadium. But at any rate, they are lost forever.
One member of the Dutch Putmans, David a son of Johannes Putman or Poutman the original immigrant, wandered down to Monmouth County, New Jersey in 1708 had a bunch of kids and then went to Hunterdon County, New Jersey in about 1732 and remained there until his death in Pluckemin, Hunterdon County on October 28, 1761. A few of the kids hung around, the rest said to heck with it and went back to New York.
A few hardy souls out of the New England gang wandered down after the Revolution, but none of our Southern folks seem to have ventured into the Garden State. The earliest guy to come down from New England was Miles Putnam (number 517 in Eben's books). Miles came down from Grafton, Vermont in the late1790s and settled in Plainfield, Somerset County, where he married and then taught school until his death in 1827. It was his family, and only his, that appear in the 1830 and 1840 New Jersey census reports.