We will report on some recent results on the large cardinal hierarchy between the first supercompact cardinal and Vopenka's Principle. We present various consistency results as well as a conjecture as for how the large-cardinal hierarchy of $\mathrm{Ultimate}-L$ looks like at these latitudes. The main result will be the consistency with very large cardinals of a new Kimchi-Magidor configuration; namely, we will present a model where every supercompact cardinal is supercompact with inaccessible target points. This answers a question of Bagaria and Magidor. This configuration is a consequence of a new axiom (named $\mathcal{A}$ ) which regards the mutual relationship between superstrong and tall cardinals. Time permitting we shall discuss the interplay between $\mathcal{A}$  and $\mathrm{Ultimate}-L$ and propose a few open questions.