I did that route in the winter and it was from west to east as well. I didn't start from the west coast, but I was rather coming from Utah, then straight down through New Mexico and Texas. So we really headed east on the southern tier from Texas and it was towards the end of January.
If you want to avoid the cold weather, I would leave more towards the end of January. But the south is not that cold at all. It was pretty comfortable, in fact. I guess it depends on the year, some winters might be colder, but I would not worry too much. I would worry more about tornados during the spring all the way from Louisiana to Florida. Especially around LA and Alabama. Make sure you follow the tornado warnings and do not camp when there is one coming. You might get a little bit cold the first month when the route takes you in higher altitudes, but as long you don't camp at the top of passes, you should sleep pretty well. Of course, you will still need some pretty warm sleeping bags (15oF - 25oF depending on how cold you tend to be) and a decent sleeping pad.
It is a very nice ride, nothing spectacular scenery-wise, but the southern hospitality is definitely a huge highlight. One thing I'd like to add: it is true that winter doesn't sound appealing, and yet I biked in really cold weather and found that it gave me some of my most beautiful biking souvenirs. Locals are much more eager to help you when the conditions are rough, and a warm meal or a cozy bed offered means so much when you have a bit of a hard time. And again, the southern USA is pretty comfortable in the winter. If you can, spend Mardi Gras in Oberlin, Louisiana! This will be the best time of your life!