You know that old saying "Don't put all your eggs in one basket." This can be true of applying at airlines to, it never fails to amaze me the number of people who will only apply at certain airlines. If you are that desperate to get out of your current job, why would you not apply everywhere (that would be a step up) and hope for the best.
Honestly the whole "pilot shortage" is not every really going to happen at the major level, its just not. As the majors retire the boomers they will shrink the airline, farming more out to regionals and code sharing with other carriers. Yes they will have to hire some but it's not going to be in the huge numbers that people are expecting. So why would you limit yourself to just one airline?
Lets play the numbers game shall we? So I just added up the number of pilots of 4 large regionals this is about 14,000 pilots (numbers from apc). Going by the numbers that people have posted for hiring I would guess that maybe (if no unforeseen circumstances) the airlines (non-regional)will hire about 4000 people next year. So you have about a 29% chance of being hired, remember your millage may vary as I only have numbers for 4 regionals and did not include hiring numbers from cargo companies.
The pilot shortage doesn't look so good when you and more than 14,000 of your closest friends are competing for 4,000 jobs. Its unfortunate but for right now it's the reality. This means that unless you are happy staying at your regional and never moving on you need to apply everywhere. Yes, everyone wants to go to Delta, the reality is they are only hiring around 1400 people this year out of more than 14,000. And before everyone freaks out, I'm not saying it's not possible I'm saying it's not probable. So unless you are happy staying at your regional (or wherever you are) apply everywhere. Other wise please don't come crying to me about how much you hate your regional but have only applied to one airline.
* regionals used were XJT, SkyWest AE and Republic
*majors hiring numbers were from DAL UAL, AA, NKS F9, SWA, JB