Well I can say of the major airline pilots I've met over the years, it seems a disproportionate amount have come from Mesa......I guess working there is motivation enough to get your resumes sent out.
If you go to Mesa, work hard, keep your nose clean, upgrade, etc, then you'll join the literally thousands of others who are qualified to apply to the majors that are hiring, but aren't receiving phone calls.
Your best bet is to approach the regional job hunt with the intent that if you get stuck there, it's a place you'd want to be. I had the choice between Air Midwest (Mesa) and ACA in 2000 - I chose the Blue Ridge team. If things go well, I could be at national airline flying Airbuses in a few years, then again, we could also be gobbled up by Mesa (yuk!).
You would also do well to go to companies like Comair, ASA, Skywest, etc..., both which offer good career opportunities for regional airline pilots. Try to lose that mentality of "I"ve just got to get 1000 PIC, then I'm outta here." Believe me, I fly with plenty of good guys, good people, many with 2+ type ratings, multiple thousand hrs PIC, and these guys can't even get interviews, what do you think you and your 1000 PIC of Mesa time will stick out. Don't let me discourage you though, keep applying and good things will occur. I suspect when hiring resumes, the only Mesa people who may feel resentment are those who chose the Freedom route. A Captain I flew with put it best: If they aren't scabs - flying for Freedom Air- they're next worst thing. I don't have a problem with the hundreds of hard-working Mesa LNPs, and I can't see any interview committee holding much against them either.
Just my 2 cents
JT