  We propose generalised $\mathcal{N}=1$ superconformal higher-spin (SCHS)
gauge multiplets of depth $t$, $\Upsilon_{\alpha(n)\dot{\alpha}(m)}^{(t)}$,
with $n\geq m \geq 1$. At the component level, for $t>2$ they contain
generalised conformal higher-spin (CHS) gauge fields with depths $t-1$, $t$ and
$t+1$. The supermultiplets with $t=1$ and $t=2$ include both ordinary and
generalised CHS gauge fields. Super-Weyl and gauge invariant actions describing
the dynamics of $\Upsilon_{\alpha(n)\dot{\alpha}(m)}^{(t)}$ on conformally-flat
superspace backgrounds are then derived. For the case $n=m=t=1$, corresponding
to the maximal-depth conformal graviton supermultiplet, we extend this action
to Bach-flat backgrounds. Models for superconformal non-gauge multiplets, which
are expected to play an important role in the Bach-flat completions of the
models for $\Upsilon^{(t)}_{\alpha(n)\dot{\alpha}(m)}$, are also provided.
Finally we show that, on Bach-flat backgrounds, requiring gauge and Weyl
invariance does not always determine a model for a CHS field uniquely.
