  By extending the recent analysis of arXiv:2203.12203 for ${\mathcal{N}}=2$
conformal supergravity in four dimensions, we define new hyper-dilaton Weyl
multiplets for five-dimensional ${\mathcal{N}}=1$, and six-dimensional
${\mathcal{N}}=(1,0)$ conformal supergravities. These are constructed by
coupling the five- and six-dimensional standard Weyl multiplets to on-shell
hypermultiplets and reinterpreting the systems as new multiplets of conformal
supergravity. In the five-dimensional case, we also construct a new
hyper-dilaton Poincar\'e supergravity by coupling to an off-shell vector
multiplet compensator. As in four dimensions, a $BF$-coupling induces a
non-trivial scalar potential for the five-dimensional dilaton that admits
AdS$_5$ vacua.
