In the Austin area you are probably too warm to do this, but Dad would make a cage out of wire and fill that with leaves around a newly planted tree to insulate it in the winter. That was in a colder climate. With all the warm weather youll have even in what is a cold winter for you, Id be concerned you would suffocate the tree.
What you might do is build up a layer of mulch around the base of the tree to protect the roots and the very bottom of the tree, but dont get carried away. Dont get too deep.
Yeah, when frost is predicted cover it some way. I wouldnt use plastic if it is touching the branches. I just dont trust it to not let the cold through to what it is touching. No real reason to feel that way, just that Im not comfortable with plastic. But more important than my feelings, if you use plastic, black especially, and leave it on so the sun can hit it a while, you might cook the tree.
I got some large flat sheets real cheap at a thrift store. Thats what I use to cover things to protect from frost.
If you can get that fig through its first winter it will probably be established enough you wont have to worry about it in following years unless you have a real cold spell, like close to single digits. But as long as you dont smother or cook the tree it doesnt hurt to throw a sheet over it in really cold weather.