You're not going to get hired by the majors unless you have family or close personal friend in management.
There are ten thousand regional pilots who have thousands of hours of TPIC and are 121 current. 80% of them have apps in at the majors. If they run out of them, there almost as many regional FO's, essentially all of whom have an ATP by now and are also current.
If you want to avoid regionals, you're going to have to break into corporate. You might be able to do that piecemeal with some freelance CFI work to fill gaps, build recency, and network.
The ATP would be a good idea.