I can't see getting the goods to New Orleans and then having to backtrack to Eastern ports. Or taking goods from the East Coast to New Orleans to send off across the Atlantic.
There were perfectly logical reasons why New Orleans or Mobile or Galveston couldn't fully compete with ports on the Atlantic. The population centers were still in the East, so shipping would be heaviest between East Coast ports and Europe.
Why it was New York that came out on top, rather than Philadelphia, Boston, Baltimore, or Norfolk is another question, but in the 19th century geography stacked the deck against the Gulf ports.
I have posted this link often before, because it opens a window on 1850s era New Orleans.
Among other things it reports that half the US cotton crop shipped from New Orleans.
Think about that.
Now look at a map of cotton growing regions and you'll see that maybe 80% of US cotton was grown in places where Gulf Coast ports were the closest -- from Galveston to Mobile & Pensacola.
Point is: it would take hundreds of 1850s era ships to transport all that cotton from Gulf Coast ports to Europe (85%) and Northern US (15%) customers.
There's no reason to suppose such ships would stop in New York outbound, but might well deliver their inbound cargoes and immigrant passengers to New York rather than, say, New Orleans or Galveston.
Note 1860 Southern railroads: