  Using covariant expansions, recent work showed that pole skipping happens in
general holographic theories with bosonic fields at frequencies
$\mathrm{i}(l_b-s) 2\pi T$, where $l_b$ is the highest integer spin in the
theory and $s$ takes all positive integer values. We revisit this formalism in
theories with gauge symmetry and upgrade the pole-skipping condition so that it
works without having to remove the gauge redundancy. We also extend the
formalism by incorporating fermions with general spins and interactions and
show that their presence generally leads to a separate tower of pole-skipping
points at frequencies $\mathrm{i}(l_f-s)2\pi T$, $l_f$ being the highest
half-integer spin in the theory and $s$ again taking all positive integer
values. We also demonstrate the practical value of this formalism using a
selection of examples with spins $0,\frac{1}{2},1,\frac{3}{2},2$.
