I've recently started getting 'PreOverdue' emails from my local library (I notice from an internet search that other libraries, both in the UK and in North America, spell it 'pre-overdue'). The word seemed incongruous to me at first, given that 'pre' means 'before' (a certain date, in this instance), and 'over' means 'beyond' (a certain time or age) -- so, two opposites together. I suppose it makes sense -- readers are being told that their books are not yet overdue, but will be in two or three days. Now that you do everything by computer in most libraries - check books out, return them, order them - and you can renew and order from your home computer, too, you never actually come into contact with a member of staff, so fines just mount up and remain unpaid. I suppose that that is behind the 'PreOverdue' policy.
Interestingly, there is not a single word in the massive OED that begins 'preover...' or 'pre-over...'.
I wrote a post on Pre words, including pre-order and pre-drink, a couple of years ago (here).