https://www.eapisfile.com/Docs/eAPIS/private_
flyers_guide.pdf page 20
short answer - You can arrive on a private flight from Bahamas ( contiguous territory or adjacent island) if you have already been admitted to the usa less than 90 days ago and be readmitted for the remainder of those days.
This seems to relate to the fact that some airports of entry can't screen you / determine your immgration status - the stamp in your passport is just a readmission of your previous entry ( a bit like a night club hand stamp... ) . But, there was a thread on here a while back about needing a B1 or other physical visa in your passport to enter if you didn't fulfil these conditions (eg flew in from say canada without having been in the us recently) and a special one time issuance of a B1 being done at the border with a $500 charge for someone who didn't have it.