  For a fixed positive integer $e$, we describe an algorithm for computing, for
all primes $p \leq X$, the mod-$p^e$ reduction of the trace of Frobenius at $p$
of a fixed hypergeometric motive over $\mathbb{Q}$ in time quasilinear in $X$.
This extends our previous work for the mod-$p$ reduction, again combining the
Beukers--Cohen--Mellit trace formula with average polynomial time techniques of
Harvey and Harvey--Sutherland; the key new ingredient is an expanded version of
Harvey's "generic prime" construction, making it possible to incorporate
certain $p$-adic transcendental functions into the computation. One of these is
the $p$-adic Gamma function, whose average polynomial time computation is an
intermediate step which may be of independent interest. We also provide an
implementation in Sage and discuss the remaining computational issues around
computing hypergeometric $L$-series.
