  We describe in detail the derivation of a superconformal off-shell
formulation of the alternative $\mathcal{N}=2$, $d=5$ ungauged supergravity of
Nishino and Rajpoot, coupled to $n$ Abelian vector multiplets, using a general
dilaton-Weyl multiplet. We generalize the vector multiplet coupling available
in the literature and show under which assumptions that the scalar manifold
reduces to the known case of SO(1,1) $\times$ SO(1,n)/SO(n). As an application
of the formalism we propose generalized vector multiplet coupled higher
curvature terms, whose construction we sketch briefly.
