All Part 121 carriers the ATP is applicable to. As for your other concerns, sounds like flight instructing is not in your best interest, look at all avenues regarding time building, (banner towing, pipeline, networking at the local FBO and seeing if anyone needs a second pilot). All of this means is you will have to bust your hump looking, no one is going to hand it to you. And the increase in flight hours may (possibility) mean increase in pay to draw pilots.
Good luck!