Typical hyperbolic surfaces have a spectral gap greater than $2\mathrm{/}9-\epsilon$

In this article, we prove that typical hyperbolic surfaces, sampled with the Weil-Petersson probability measure, have a spectral gap at least $2\mathrm{/}9-\epsilon$. This is an intermediate result on the way to our proof of the optimal spectral gap $1\mathrm{/}4-\epsilon$, building on the results of the first part of this series. A significant part of the proof is an explicit inclusion-exclusion argument to exclude tangles at the level of precision $1\mathrm{/}g$.