I will first give a brief survey of some previous results with Sutton, in which we found a large family of decomposable Specht modules for the Hecke algebra of type B indexed by `bihooks'. We conjectured that outside of some degenerate cases, our family gave all decomposable Specht modules indexed by bihooks. There, our methods largely relied on some hands-on computation with Specht modules, working in the framework of cyclotomic KLR algebras. I will then move on to discussing a new project with Muth and Sutton, in which we have studied the structure of these Specht modules. By transporting the problem to one for Schur algebras via a Morita equivalence of Kleshchev and Muth, we are able to show that in most characteristics, these Specht modules are in fact semisimple, and give all composition factors (including their grading shifts). In some other small characteristics, we can explicitly determine the structure, including some in which the modules are `almost semisimple'. I will present this story, with some running examples that will help the audience keep track of what's going on.