  Let $\mathcal{T}$ be a Krull-Schmidt, Hom-finite triangulated category with
suspension functor $[1]$. Let $R$ be a basic rigid object, $\Gamma$ the
endomorphism algebra of $R$, and $\operatorname{\mathsf{pr}}(R)\subseteq
\mathcal{T}$ the subcategory of objects finitely presented by $R$. We
investigate the relative rigid objects, \ie $R[1]$-rigid objects of
$\mathcal{T}$. Our main results show that the $R[1]$-rigid objects in
$\operatorname{\mathsf{pr}}(R)$ are in bijection with $\tau$-rigid
$\Gamma$-modules, and the maximal $R[1]$-rigid objects with respect to
$\operatorname{\mathsf{pr}}(R)$ are in bijection with support $\tau$-tilting
$\Gamma$-modules. We also show that various previously known bijections
involving support $\tau$-tilting modules are recovered under respective
assumptions.
