#2018-383 (p. 15), but also for example #2021-633 or #2022-880 (p. 3), uses for supersingularity testing of a curve E_A : y^2 = x^3 + Ax^2 + x that if you find a rational point with order N > 4 \cdot \sqrt{p} and N \ | \ p+1 then the curve E_A must be supersingular, as \#E_A(\mathbb{F}_p) = p + 1 is then the only possible value in the Hasse interval [p + 1 - 2 \sqrt{p}] < \#E_A(\mathbb{F}_p) < [p + 1 + 2 \sqrt{p}].

But surely this holds already for N > 2 \cdot \sqrt{p} or am I missing something?

I.e. N \cdot \alpha = p+1 together with N > 2\cdot \sqrt{p} implies both N \cdot (\alpha - 1) < p + 1 - 2 \sqrt{p} and N \cdot (\alpha + 1) > p + 1 + 2 \sqrt{p}