In the old days, the better airlines would give you a full astronaut medical, at least an all day affair, and hold you to their own standards which would make NASA envious. They presumably would have wanted to see documents related to any lingering conditions. But over the years they incurred some legal liability and also had to cut costs as the industry evolved post-deregulation.
These days most majors and I think all regionals simply make a copy of your FAA 1C. Saves them money, avoids labor lawsuits, and lets the FAA be liable for your medical suitability.
A couple airlines still do an exam but it's only after a CJO, and they only verify that you meet 1C standards (in case you got your medical from a Santa Claus AME). Pretty sure those airlines do a normal medical, nothing more nothing less, with an actual AME so you do have to fill to fill out the form and cannot lie (no law against lying to employer, but can't lie to an AME). In that case you might want to have any approval letters you previously got from the FAA, or possibly other medical docs. I'd bring what you have but don't get them out unless asked. Again, only a couple of the legacies will even bother with an exam.
That's for the US. Outside the US, whole different story of course.