U.S. Mail coming in and out of the country was required to go through the Port of New York. Packets were being paid to take mail to Europe in otherwise empty ships, where they loaded manufactures and transported them home at the going rate.
They figured out they could take Southern cotton (and other commodities) at rock-bottom prices with them and they did so. It's because of this that goods were shipped to New York, unloaded on the docks and re-loaded onto ships whose passages to Europe were being subsidized by the Postal Service.
You might be confusing ocean steamer packet service with coastal packets, which did routinely haul cotton to northern ports, and as it was discovered, despite having a relatively shallow draught did a fair job in transAtlantic service too.
If you have examples of steamer mail packets under contract with the Navy of Post Office hauling cotton, I would be interested in them.
Even if mail packets had hauled cotton out of NYC, in the cotton shipping season (Oct - March) of 1859-60 for example, there were only four lines hauling mail - the Cunard Line, the Vanderbilt Line, the Havre Line and the Inman Line and a combined total between all four of only 44 departures - 34 to Liverpool and 10 to le Havre. That doesn't put much of a dent in the approx. 4.5 million bales of cotton exported from the US.