  Vazirani and the author \cite{BV} gave a new interpretation of what we called
$\ell$-partitions, also known as $(\ell,0)$-Carter partitions. The primary
interpretation of such a partition $\lambda$ is that it corresponds to a Specht
module $S^{\lambda}$ which remains irreducible over the finite Hecke algebra
$H_n(q)$ when $q$ is specialized to a primitive $\ell^{th}$ root of unity. To
accomplish this we relied heavily on the description of such a partition in
terms of its hook lengths, a condition provided by James and Mathas. In this
paper, I use a new description of the crystal $reg_\ell$ which helps extend
previous results to all $(\ell,0)$-JM partitions (similar to $(\ell,0)$-Carter
partitions, but not necessarily $\ell$-regular), by using an analogous
condition for hook lengths which was proven by work of Lyle and Fayers.
