Hi ! I work for Southwest Airlines (disclaimer: anything and everything stated here is the opinion of the individual and not in way acting as a spokesperson/official representative of Southwest Airlines) Gotta CYA, ya know? So in other words I am not officially representing Southwest, but I am an operations agent, who is the person you hand your boarding card to when you board the airplane.
The suggestions mentioned above are all good ones, and I will mostly repeat here:
Options:
1. Buy seat for the dress (absolute safest method)
2. Put dress in overhead (you're getting it steamed once you get there anyway, right?)
3. This one is super-iffy. SOME Southwest planes have a (extremely small) "closet" ... okay, more like "small bar you could put a hanger or two on" in the flight deck. You can ask the operations agent (or the customer service agent, basically anyone at your gate, but you specifically want to seek out the one nearest the actual boarding door) if the Captain would mind if you hung your dress up front to keep it safe. There is never any guarantee of this, the space may or may not exist depending on the plane, and it is completely up to the Captain's discretion beyond that. Always plan your back up to be putting your dress in the overhead.
Someone mentioned above to pay the fee for "Early Bird Check-in" which will get you on the plane sooner and therefore have more overhead space available. Southwest also has a Business Select option which gets you through the first class line at TSA, 12x Rapid Reward Points, position A1-A15 in boarding line, and a free cocktail or energy beverage on the plane.
ONE more thing!!!! This depends on where you are flying out of, how busy it is, and a few other factors. Anyway, tell everyone you are flying out to get married! Tell the ticket agent, the customer service agent at the gate, and (if you see them early enough) the agent at the boarding door. Be sure not to say it in an assuming or expecting manner. Just work it in causally (asking about where to store your wedding dress on board is the perfect time) IF there is enough time and IF you are flying out a station that has such things ... you just miiiiight get a bubbly surprise on board. No guarantees, but it's lovely when it can be pulled off.
Whew. Any other questions about airlines or Southwest specifically, feel free to ask