Indeed there are many out there with the minimum qualifications you speak of, who are finding it very tough to find a break into the airlines. There are threads constantly about job prospects and 'who's recruiting' etc and it is clear that the first break is (and always has been) extremely difficult to secure.
FATPL holders come in many shapes and sizes and, just like in other walks of life, can take varying amounts of knockbacks before they throw in the towel. The one thing that seems to really stand out now is the perception that if you don't get your 'big break' within a few months of qualifying then the market is somehow unfair. In years gone by many pilots here in the UK spent considerably longer than 'several months' building their experience and enhancing their CV before finally getting the first break. I suspect this expectation of early success comes from seeing a large number of low houred pilots being taken on through schemes like the CTC ATP program and possibly also because of the increasingly large sums of money that young pilots get into debt for and have to start paying back sooner rather than later.
From personal experience of training back in 1997-1999, all except one pilot who qualified with me have airline jobs. The other one is an FI. Some were lucky and got a job straight out of training, others gained an airline job through the CTC scheme, some went air taxi or FI and some spent 12 months or so trying to get in with turbo-prop outfits before getting their break. Off the top of my head these are the airlines that I know some of these guys work for;
Britannia (x2)
Thomas Cook (x2)
EasyJet
Gold Air
Aurigny
BACE
Aer Lingus
BMed
So you can see that what it comes down to is persistence and never giving up. Don't expect a job straight away, but do everything to give yourself the best chance of getting one straight away. My best advice would be at all costs to not end up 12 or 24 months after qualifying with no more hours on your now slightly less shiny CPL...........think about your backup plan should the inevitable happen (ie you don't get a job flying a 747-400 one week after qualifying!!)
Good luck.
PP