Intermediate $\mathsf{NP}$-complete problems?
This is a partial answer that only gives a candidate intermediate $NP$-complete problem.

$k$-Equal Sum Subsets problem: Given a multiset of $n$ positive integers $A = \{a_1,...,a_n\}$, are there $k$ nonempty disjoint subsets $S_1,...,S_k ⊆ \{a_1,...,a_n\}$ such that $sum(S_1) = ... = sum(S_k)$?

The problem is weakly $NP$-complete when $k= O(1)$ and therefore has pseudo-polynomial time algorithm for any fixed constant integer $k &gt; 2$. However, it becomes strongly $NP$-complete when the number of equal sum subsets $k= \Omega(n)$.

If $k= \omega(1)$ and $k= O(\log n)$ then $k$-Equal Sum Subsets problem is a candidate intermediate $NP$-complete problem (as described in the question). This problem is neither known to have a pseudo-polynomial time algorithm nor proven to be $NP$-complete in the strong sense.

Reference:

CIELIEBAK, EIDENBENZ, PAGOURTZIS, and SCHLUDE, ON THE COMPLEXITY OF VARIATIONS OF EQUAL SUM SUBSETS, Nordic Journal of Computing 14(2008), 151–172