We study the infinite generation in the homotopy groups of the group of diffeomorphisms of $S^1 times D^{2n-1}$, for $2n geq 6$, in a range of degrees up to $n-2$. Our analysis relies on understanding the homotopy fibre of a linearisation map from the plus-construction of the classifying space of certain space of self-embeddings of stabilisations of this manifold to a form of Hermitian $K$-theory of the integral group ring of $pi_1(S^1)$. We also show that these homotopy groups vanish rationally.
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