Yes you do. The Big 3 have even hired 60 year olds in recent times. Do whatever it takes to get to 1500 hours and an ATP as soon as possible then get on with a regional airline, a fractional, or an LCC/ULCC. Áll of those have hired 1500 hour pilots in recent times.

If your goal is to get to a “major” (I think you’re actually referring to the Big 3, SWA, UPS or FedEx), then I’d prioritize an LCC/ULCC over a regional airline, and a regional airline over a fractional. Part 121 time is preferable over time at the fractionals.

The aviation world is your oyster at the present time but time is of the essence. The good times won’t last forever. It’s an industry full of up and down cycles.