  Previously, the last two authors found large families of decomposable Specht
modules labelled by bihooks, over the Iwahori--Hecke algebra of type $B$. In
most cases we conjectured that these were the only decomposable Specht modules
labelled by bihooks, proving it in some instances. Inspired by a recent
semisimplicity result of Bowman, Bessenrodt and the third author, we look back
at our decomposable Specht modules and show that they are often either
semisimple, or very close to being so. We obtain their exact structure and
composition factors in these cases. In the process, we determine the graded
decomposition numbers for almost all of the decomposable Specht modules indexed
by bihooks.
