  Daisy graphs of a rooted graph $G$ with the root $r$ were recently introduced
as a generalization of daisy cubes, a class of isometric subgraphs of
hypercubes. In this paper we first solve the problem posed in
\cite{Taranenko2020} and characterize rooted graphs $G$ with the root $r$ for
which all daisy graphs of $G$ with respect to $r$ are isometric in $G$. We
continue the investigation of daisy graphs $G$ (generated by $X$) of a Hamming
graph $H$ and characterize those daisy graphs generated by $X$ of cardinality 2
that are isometric in $H$. Finally, we give a characterization of isometric
daisy graphs of a Hamming graph $K_{k_1}\Box \ldots \Box K_{k_n}$ with respect
to $0^n$ in terms of an expansion procedure.
