  We present a consistent way of coupling three-dimensional Maxwell
Chern-Simons gravity theory with massless spin-$\frac{5}{2}$ gauge fields. We
first introduce the simplest hyper-Maxwell Chern-Simons gravity containing two
massless spin-2 fields coupled with a massless Majorana fermion of
spin-$\frac{5}{2}$ and whose novel underlying superalgebra is explicitly
constructed. Then, we present three alternative hypersymmetric extensions of
the Maxwell algebra which are shown to emerge from the In\"on\"u-Wigner
contraction procedure of precise combinations of the
$\mathfrak{osp}\left(1|4\right)$ and the $\mathfrak{sp}\left(4\right)$
algebras. This allow us to construct distinct types of hyper-Maxwell
Chern-Simons theories that extend to include generically interacting
non-propagating spin-$4$ fields accompanied by one or two spin-$\frac{5}{2}$
gauge fields.
