Summary: 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 address a problem posed in [\textit{A. Taranenko}, Eur. J. Comb. 85, Article ID 103058, 10 p. (2020; Zbl 1433.05105)] and characterize rooted graphs \(G\) with the root \(r\) for which all daisy graphs of \(G\) with respect to \(r\) are isometric in \(G\), assuming the graph \(G\) satisfies the rooted triangle condition. We continue the investigation of daisy graphs \(G\) (generated by \(X)\) of a Hamming graph \(\mathcal{H}\) and characterize those daisy graphs generated by \(X\) of cardinality 2 that are isometric in \(\mathcal{H} \). Finally, we give a characterization of isometric daisy graphs of a Hamming graph \(K_{k_1}\Box \cdots \Box K_{k_n}\) with respect to \(0^n\) in terms of an expansion procedure.