Natural candidates for the hierarchy inside NPI
Group Isomorphism $\leq_m$ Graph Isomorphism $\leq_m$ Ring Isomorphism. Also Integer Factoring $\leq_m$ Ring Isomorphism [Kayal and Saxena]. Also Graph Automorphism $\leq_m$ Graph Isomorphism.

Not only are there no known reductions the other way, but there is provably no 
$\mathsf{AC}^0$-reduction from Graph Iso to Group Iso [Chattopadhyay, Toran, and Wagner]. 

Note that a reduction from Ring Isomorphism to Graph Isomorphism would also provide a reduction from Integer Factoring to Graph Isomorphism. To me, such a reduction would be surprising though perhaps not shocking.

(For Graph Automorphism vs Graph Isomorphism, their counting versions are known to be equivalent to one another and equivalent to deciding Graph Isomorphism. However, that's not necessarily saying much, as the counting version of bipartite matching is equivalent to the counting version of SAT.)

I don't think there is a real consensus as to which, if any, of these are actually in $\mathsf{P}$. If any of these problems is $\mathsf{NP}$-complete then $\mathsf{PH}$ collapses to the second level. If factoring is $\mathsf{NP}$-complete, then it collapses to the first level, i.e. $\mathsf{NP} = \mathsf{coNP}$.

Also, I seem to recall that using techniques similar to Ladner one can show that any countable partial ordering can be embedded in the ordering $\leq_{m}$ on the problems in $\mathsf{NP}$ (so it's not just a hierarchy, but an arbitrarily complicated countable partial order).