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While a mat might make his landing softer, given the breed they are prone to disc issues which can cause problems when then jump off of furniture.
I would rather see you train him to use the step at all times, especially since he is getting older. You can train this by picking up some bed rails at a used store such as salvation army or even a yard sale, and put them up on either side of the steps, then that will kind of force him to use the steps. At that point you want to put a command to the steps, and praise and reward with something special each time he uses the steps rather than jumping off.