If I were going to add extra chicken, I'd add it raw. It'll boil nicely in 1.5 hours or so at a simmer.

And-- it may just be where I come from, or it might just be my personal taste, but I rarely skim fat from anything, unless there is just a whole lot way too much of it. I do drain mince when cooking with it, but I don't skim fat off the top of chicken soup-- I stir it in.