$ \mathcal{C}^{(1)} $ is the Pauli group and all elements of the Pauli group are monomial matrices. Thus the monomial matrices in the first level of the Clifford hierarchy are exactly all the Pauli matrices.
Lemma: Suppose that $ U,V $ are unitaries such that
$$
U=VC
$$
for some Clifford gate $ C $. Then $ U \in \mathcal{C}^{(k)} $ if and only if $ V \in \mathcal{C}^{(k)} $.
Proof. $ U \in \mathcal{C}^{(k)} $ iff
$$
UPU^{-1} \in \mathcal{C}^{(k-1)}
$$
for all $ P \in \mathcal{C}^{(1)} $ iff
$$
(VC)P(VC)^{-1}=VCPC^{-1}V^{-1} \in \mathcal{C}^{(k-1)}
$$
for all $ P \in \mathcal{C}^{(1)} $
iff
$$
VP'V^{-1} \in \mathcal{C}^{(k-1)}
$$
for all $ P' \in \mathcal{C}^{(1)} $ (recall conjugation by a Clifford gate is an automorphism of the Pauli group, by definition) which is true iff $ V \in \mathcal{C}^{(k)} $
Recall that every monomial matrix $ M $ can be written as
$$
M=DP
$$
for $ D $ a diagonal matrix and $ P $ a permutation matrix (to get $ P $ just replace all the nonzero entries of $ M $ by $ 1 $ and to get $ D $ just take all the nonzero entries row by row and list them down the the diagonal).
So any monomial matrix whose corresponding permutation matrix $ P $ is in the Clifford group and whose corresponding diagonal matrix $ D $ is in the Clifford hierarchy will be in the Clifford hierarchy . And the diagonal matrix $ D $ is in the Clifford hierarchy iff it has a certain form in terms of $ 2^k $ roots of unity see
https://arxiv.org/abs/1608.06596
unclear what kind of permutation matrices show up in the Clifford group/ Clifford hierarchy. For example Toffoli shows up in third level but not second level. Recall that Toffoli is $ CCX $. So this is an example of the general fact that $ C^kX $ is in the $ k+1 $ level of the Clifford hierarchy but not in the $ k $ level of the Clifford hierarchy.
