You want to add 6 to 10 chicks or pullets to your current flock. Your existing coop section is probably 5' x 10'. In Chattanooga you should have weather where the chickens can be outside practically every day of the year. Your run is not predator proof with that bird netting but the pen can be with dig protection. I don't know how many chickens you currently have but you probably have enough coop space if you can leave the door to that pen open 24/7 or at least when they are awake. Tight sleeping space, especially when integrating, could be an issue. Integration is harder when you don't have extra room. I would not take any more space away from that 5x10 coop. I'd feel a lot better about adding 6 more chicks instead of 10. Your room in the coop is going to be tight for sleeping even if you can use that pen for when they are awake. Your 16x30 run exceeds the suggestion you often see in here of 10 square feet per chicken but my enclosed run is 12' x 32'. It looks pretty crowded when I have 20 chickens in there and I'm integrating.
For baby chicks you could build a brooder in that storage area with wire separating the coop and brooder so they can see each other. I'm assuming your "pullets' does not mean POL pullets. That brooder in the storage area can work but it is not what I'd do.
I would build a "grow-out" coop in that 16x30 run and put a predator proof run around it. You'll need to run electricity to it for heat. For 10 baby chicks a 4' x 4' coop would be big enough. Plywood sheets and some paneling comes in 4' x 8' dimensions and the cheaper lumber is usually in 8' lengths so a 4x4 is reasonable to build. I'd put roosts in but no nest though a nest won't hurt if you use it to isolate hens later. I made mine 4' x 8' and set it outside of my 12x 32 run with a pop door to a 8x12 run section fenced off in the run but my circumstances are different to yours. I needed more room, you don't need that much room in the little run. I elevated mine a bit and covered the bottom with 1/2" hardware cloth so the chick's poop drops through, keeps it dry and clean. When you are raising baby chicks in it, put a piece of plywood or paneling on the floor to keep heat in, you can tilt it and rake it off to clean. With that wire floor it can serve as a broody buster if you don't have baby chicks. You can use it as a hospital or to isolate a chicken if you need to. Having a separate facility attached to the main one has come in handy in many different ways. If you build a brooder in the storage area you can build it as a broody buster or as a hospital too.
If your "pullets" does mean POL pullets I'd make the small coop 4' x 8'.