If you are lucky, and you look carefully right above the entry of your ancestor on a ship's manifest, you just might find the file number and date of naturalization written like notes. There were years when they needed to have the documentation for the immigrant, and I believe during these years they made notes right on the manifest. They are small and often numbers that one overlooks, they are usually written above the entry, kind of squeezed in there. Not above the name but a little further across kind of above where the occupation is or even futrther across on the first page.
Another thing you might find, is that for women who traveled under their own names, you will find that they have crossed out that surname and written above is her married name. This does not happen that often but once in a while.
I do transcribing of manifests and have found these file numbers and dates many times. If you want to know what they look like just take a look at the attached, passenger #20 D'Agostino, Vincenzo. You can see the notations.