Only 1000 PIC after 3 yrs as a T-38 FAIP? No IP time? If there's a story behind that, make darn sure you chat with someone with an HR background about how to consistently present whatever happened.

As for what to do now... You meet the min requirements I think so you might be fine almost anywhere getting 121 time. A few of my mil friends with poor recency or low PIC time seem to have ended up at skywest, with at least one of them sticking it out a few years to upgrade and start building PIC time.

From the website - Flight Experience: 2,500 hours total or 1,500 hours Turbine total. Additionally, a minimum of 1,000 hours in Turbine aircraft as the Pilot in Command* is preferred. Southwest considers only Pilot time in fixed-wing aircraft.

If you have that, as far as I can tell it's recency and recommendations that gets you the interview sooner rather than later. Based on anecdotal evidence I've seen, just one year of active flying may be all you really need especially if you have any internal recommendations to help out.

Also, don't wait to apply until you think you meet the super secret minimums. Apply immediately. Keep your application, resume, and recommendations current. Re-apply every time the application window opens, and make sure your email account isn't sending SWA HR email to the spam folder since that seems to be an issue for some people.

Also also, don't wait until you have an invitation to do your HR prep. There are a few people and groups out there who have a good reputation for making sure you're ready for the SWA interview process, so make sure you get that done early. More specifically, if possible you should get your resume and application reviewed before your first application window to make sure you don't have any unintentional errors that when corrected, might raise integrity questions.