My understanding of US law until around 1914 was that you didn’t need a passport, work permit or any other official documentation to enter the US. You could be refused entry on the grounds of health, criminal record (though how that would ever be known in most cases is beyond me) and because you might fall a charge on public funds. The percentage actually refused entry was very low 0.2% or less I seem to recall, so the need to jump ship or use some form of deception would seem to me to be pretty low. I would have thought that most people just took their chances. It was all a fairly simple process, and though no doubt people worried, the risk of being sent back was pretty low.
The ones who might have reason to practice some form of illegal migration, would probably just be that small number who had been sent back, and who decided to have another go. I would have thought they would have been keen to avoid going back to the same port of entry that they used on the first time. Perhaps also a land border crossing from Canada would be seen as helpful because if refused admission again there, you didn’t have to buy another ticket across the Atlantic.
I’d be surprised if there was any methodology that led to would be migrants choosing one UK or Irish port of departure over another. After all you still faced the same officials in the US regardless of where you had come from. Liverpool was always the main port of departure simply because there were far more sailings from there then from Irish ports. I think the port of departure was simply a matter of availability of places and the cost of the fare. (For a long time there was fierce competition for the trade, and many ships agents threw the cost of passage from Ireland to Liverpool in free as part of the deal).
Jumping ship was behaviour more associated with members of the ships crew, than with the passengers who were supposedly supervised until they cleared immigration. The crew though could usually go ashore unsupervised.
However this is all just my opinion. I don’t know of any specific cases, which is really what you are enquiring about.