As it turns out, I probably don't need to take an earlier flight this time (except to insure against the possibility of my booked flight being delayed), but the information here is good to have.

My ticket isn't a "stand-by ticket," it's for a specific flight, but it's one of the economy fares (SWA calls it a "Wanna Getaway" fare) and their policy is that those are not eligible for going stand-by on another flight. However, at least once they've put me on a LATER flight - which I'd think qualifies as stand-by - when I missed my ticketed flight, and I didn't have to "upgrade" my ticket.   So I do think it's possible they'd do the same if I got there early and asked to be put on an earlier, emptier flight.

There are two ways to go about this:
1. Inquire at the ticketing/check-in counter before going thru security.   Ask the agent if they can change the ticket to an earlier flight (and what, if anything, that would cost).
2. Check in - online or at the airport counter - for the ticketed flight, go through security and go to the gate where the next flight for that airline is leaving from. Ask the agent there if you can get on that flight if there's room.   This would involve the same clerical work that's involved when you miss a flight or volunteer to be bumped and they re-route you to your destination.

I think the second option is more likely to work.    (If you've checked bags, they'll just arrive with the flight you were originally ticketed for.)
Of course, if you try option #1 first, you have two shots (two different ticketing agents) at the goal.   And if an agent changes your ticket before you check your bags, they'll be ont he same flight as you.

In this case I'm not concerned with checked baggage.