I started a new lawn about a month ago. I live in northern Jersey and the
backyard is about 50% shady. I used Scott's fall mix.

While I do have a lot of good grass...I also have a lot of bald patches (a
few of the bald patches are due to my dog digging it up when I'm not
paying
attention to him).

I assume that being it's getting cold outside I need to wait to the spring
to put more seed down? If this is correct, what month should I do this
in?

I'd go ahead and toss it down now anyway -- worst case scenario, you have to
spend 15 bucks on another bag of grass seed next spring. I did the same
as you and started filling patches about a week ago -- it's doing just fine.
FWIW, I've never been much for the extra special patching products -- just
good ole grass seed always does the trick for me.

James
Zone 6, SW Ohio