  We will use the covariant superform approach to develop a new density formula
for $\mathcal{N}=2$ conformal supergravity which is based on a fermionic
multiplet whose lowest component is a dimension-5/2 spinor. We will show that
this density formula admits an embedding of the real scalar multiplet of
[arXiv:1712.02309]. Upon using the embedding of the tensor multiplet into the
real scalar multiplet, we will construct a new higher derivative action of the
tensor multiplet in $\mathcal{N}=2$ conformal supergravity.
