Especially living in TX, I would not go 0w20. I wouldn't even go 5w20. Depending on how many miles you have on your 3.8, I would run 5w30, 10w30 or even 10w40. 5w20 was spec'd on the 3.8 to improve fuel consumption. Thinner oil flows better and takes less energy to pump (think of drinking milk through a straw vs drinking a milkshake through a straw).
So if anything, stick with the factory recommended, maybe go thicker, but I certainly would not go thinner.
2014 Mustang GT Track Pack
2015 WK2 Limited 4x4