  We return to some past studies of hyperkloosterman sums ([9,10]) via $p$-adic
cohomology with an aim to improve earlier results. In particular, we work here
with Dwork's $\theta_\infty$-splitting function and a better choice of basis
for cohomology. To a large extent, we are guided to this choice of basis by our
recent work on the $p$-integrality of coefficients of $A$-hypergeometric
series[3]. In the earlier work, congruence estimates were limited to $p>n+2$.
We are here able to remove all characteristic restrictions from earlier
results.
