I'm a civilian, but go ahead and apply for a job at the majors and wait for the call while flying for regionals. You'll get plenty of flight time plus lots of experience in the 121 world as well as increase your list of contacts. The thing is to also upgrade ASAP.

You've got a rare opportunity to have the best of both worlds for your résumé at both jobs. See where you want to be in 5 years. It's been a proven decision maker for me.

Good luck.