Let the sides of the right triangle be $a, b, \sqrt{a^2+b^2}$. Then, we have that we want to maximize $\sqrt{(a + 1)^2 + (b + 1)^2} - \sqrt{a^2 + b^2}$. We can use triangle inequality on this. Set points at $(0, 0), (a, b), (a+1, b+1)$. Then, using this geometric interpretation, we can see that that is maximized at $\sqrt{2}$, so the answer to the question is no.