Persuading a school to increase its offer is easier than many people realize. The trick is knowing how — and when — to ask.

It’s that terrifying time for seniors — high school seniors, that is: the nail-biting moment when millions of them find out whether they’ve been accepted or rejected by their college of choice. But if that in-or-out verdict appears final, there’s another one that’s anything but definitive — namely, a college’s offer of financial help to parent and child.

At a time when demand for college aid is soaring — applications for federal assistance have increased by 59 percent since 2006 — 07, according to the U.S. Department of Education and FinAid.org — appeals of award packages are also on the rise. Some colleges say requests for reconsideration are up as much as 30 percent over the past three years; this is forcing administrators to enter into sensitive financial negotiations and even renegotiations (yes, you can appeal an appeal). And the back-and-forth involves not just newly accepted students but also those at the tail end of their campus experience.