In exceptional times ('98-2001 was one such), airlines have indeed approached students while still on courses, but those who had graduated some time previously and were marking time, and had perhaps lost currency, were still finding things difficult.
I think we have interesting times ahead; my contacts report that BA is struggling to fill its latest direct-entry quota for type-rated pilots (though that process may be suspended due to a pension dispute), easyJet, Ryanair, Britannia, Virgin Atlantic, FlyBe and BACX are all recruiting in UK - and not finding things easy; most of the Middle and Far East airlines are also recruiting, and the pool of experienced and type-rated pilots is fast running out. That means that many of these jobs will be filled by poaching from other airlines (which has positive ramifications for pay and conditions), and the rest will have to be filled by inexperienced, unrated pilots.
That means you guys!
Scroggs