I agree with Rufus, patch the existing hole and go with a hole higher up. It's not worth the trouble to connect it low only to find out you will get smoke back into the room. I doubt that you will find an inspector that will approve the dip in the pipe, before it goes up. You will just be creating a lot of smoke and creosote.
Do you also realize that you must use "Class A" insulated pipe, which has a 2" clearance to combustible materials, such as your drywall and wood structure behind the drywall. Then per "code" you will need to use a stainless steel liner up through the existing chimney, if you don't have one already.
Technically, once you install the pipe up higher into a tee connector, no heat or smoke can come down to the existing hole, so a metal plate, screws, and sealant and them mud in the rest, to patch up the hole.
I can't tell if that is a 6" or 8" exhaust, but that will be the size of the liner you will need to run up through the chimney.