You might need the round-to-oval adapter. It depends on whether that's a common oval duct or a round duct that has merely been "squished" a bit to fit inside a 2x4 wall. I can't tell which, from the photo.

As for how to hook up the rest? That depends a lot on whether your existing dryer duct lines up with (or extremely close to) the wall duct, when the dryer is pushed into place. If it lines up, you may be able to use something like an adjustable rigid-metal elbow, right off the dryer, going up to the wall duct (or adapter). If it doesn't line up, then what would be best is an elbow right off the dryer, then a length of straight rigid duct, then another elbow or two leading to the wall duct (or adapter). As an alternative, you could use a metal semi-rigid duct, going right from the dryer to the wall duct (or adapter).

I can't be certain, but it appears that the duct in your first photo is a length of semi-rigid duct with a small piece of rigid duct that is either an oval adapter or has been squished to fit inside the oval. If so, you can just connect the oval piece to the wall duct and tighten the clamp, then connect the other end to the dryer and tighten the clamp. (You don't need to "stretch" the semi-rigid duct any more than is needed for it to reach.) Just be sure that you don't crush the duct, when you push the dryer back against the wall.