just because the car can run after it's jumped doesn't negate the alt as an option. Sometimes an alt can go bad but still put out enough juice that it seems ok for a while. I just replaced one for a friend 2 weeks ago that was putting out 12 volts. It should put out 13.5-15 volts. Your battery could also be bad, even though it's new - running a battery dead a few times is often all it takes to kill it. You could also have a short or bad ground somewhere.
I'd start by checking the alt. Jump the car, and with it running (and disconnected from the other car), use a multimeter across the battery terminals to check for volts. If it's less than 13.5, your alt is probably junk. There are sometimes situations where the alt seems fine at an idle, but isn't fine under a load.
Since this is happening overnight, there's a decent chance it's a ground thing, just don't count other things out yet.