We consider the problem of characterising the generic rigidity of bar-joint frameworks in $mathbb{R}^d$ in which each vertex is constrained to lie in a given affine subspace. The special case when $d=2$ was previously solved by I. Streinu and L. Theran in 2010 and the case when each vertex is constrained to lie in an affine subspace of dimension $t$, and $dgeq t(t-1)$ was solved by Cruickshank, Guler and the first two authors in 2019. We extend the latter result by showing that the given characterisation holds whenever $dgeq 2t$.
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