  We use higher Coleman theory to construct a new $p$-adic $L$-function for
$\text{GSp}_4 \times \text{GL}_2$. While previous works by the first author,
Pilloni, Skinner and Zerbes had considered the $p$-adic variation of classes in
the $H^2$ of Shimura varieties for $\text{GSp}_4$, in this note we explore the
interpolation of classes in the $H^1$, which allows us to access to a different
range of weights. Further, we show an interpolation property in terms of
complex $L$-values using the algebraicity results established in previous work
by the authors.
