Is the collapse of $PH$ known to extend downward to classes in-between its levels?
I don't have a good answer, but in the spirit of complexity, I have some answers which suggest that a good answer may be hard to come by :).


Note that the generalized version of Ladner's Theorem implies that there are infinitely many poly-time degrees strictly in between $\mathsf{\Sigma_i P}$ and any poly-time degree strictly above it. In particular, if the hierarchy collapses to the $i+1$-st level but not the $i$-th, then there are infinitely many p-degrees between $\mathsf{\Sigma_i P}$ and $\mathsf{\Sigma_{i+1} P} \cap \mathsf{\Pi_{i+1} P}=\mathsf{\Sigma_{i+1} P}$.
If I recall correctly, constructing an oracle for which $\mathsf{PH}$ looks like the arithmetic hierarchy is still an open problem. By "looks like the arithmetic hierarchy," I mean that $\mathsf{PH}$ does not collapse, and $\mathsf{\Sigma_{k} P} \cup \mathsf{\Pi_k P} \subsetneq \mathsf{\Delta_{k+1} P} = \mathsf{\Sigma_{k+1} P} \cap \mathsf{\Pi_{k+1} P}$ for all $k$. This at least suggests that the answer to your question may not be known.
Ker-I Ko gives oracles in which he separates the levels of $\mathsf{BPH}$ from those of $\mathsf{PH}$. As these two hierarchies intertwine one another, this gives you at least some information about relativizable collapses of problems between the levels of $\mathsf{PH}$.
This next reference is the wrong direction, but you might also be interested in the result and its techniques. Chang and Kadin showed that if the Boolean hierarchy (which lives entirely below the second level of $\mathsf{PH}$, but extends $\mathsf{DP}$ to a whole hierarchy) collapses to its $k$-th level then $\mathsf{PH}$ collapses to the $k$-th level of the Boolean hierarchy over $\mathsf{\Sigma_2 P}$.