Hi Christian
eAPIS is a border control measure and is only needed at point of entry and point of departure from US territory. Internal US sectors don't count and CBP aren't interested.
Based on what you have said, you do need a TSA Waiver - broadly equivalent of a landing permit - which lists all locations you are operating to (including before and after USA), crew and aircraft details.
Carry copies of both with your crew docs.
The only other document we routinely give our US flights is a finance form that declares importing >$10k, which keeps CBP happy.