The lug wrench may have been out if he thought about using the car's jack to try to get something under the wheels for traction.
The lug wrench is usually stored with the jack and used to operate the jack.
Exactly; that is what I was thinking, that Matthew spent considerable time trying to free his vehicle. He probably tried using the jack, and he probably contemplated using the wrench to get the back tire off, so he could build support beneath the wheel with rocks and regain traction (rear-wheel drive). In the photo, one can even see tire rub marks on the rock as some contact between the rock and the tire was achieved and the car was then put in gear.
If anyone questions whether it was someone other than Matthew who drove the vehicle there should consider the time and effort invested into attempting to free the vehicle. A criminal would not want to be seen around the car longer than needed. And even if that criminal was trying to free the car, what was the point? Drive it the rest of the 10 yards to the large bolder just ahead? Drive it back out? Dump it over the edge? The vehicle could have been dumped way earlier, not a whole mile from the main road. Matthew's phone activity also reinforces the theory that Matthew was not in some peril with an unknown criminal at Rosa's Lookout.