  We provide a new bound on the maximum degree of the Jones polynomial of a
positive link with second Jones coefficient equal to $\pm 1$ or $\pm 2$. This
builds upon the result of our previous work, in which we found such a bound for
positive fibered links. We also show that each of these three bounds obstruct
positivity for infinitely many almost-positive diagrams, allowing us to
classify infinitely many knots as almost-positive.
