A temporal graph G = (G_1, G_2, ..., G_T) is a graph represented by a sequence of T graphs over a common set of vertices, such that at the i-th time step only the edge set E_i is active. The temporal graph exploration problem asks for a shortest temporal walk on some temporal graph visiting every vertex. We show that temporal graphs with n vertices can be explored in O(k n^{1.5} log n) days if the underlying graph has treewidth k and in O(n^{1.75} log n) days if the underlying graph is planar. Furthermore, we show that any temporal graph whose underlying graph is a cycle with k chords can be explored in at most 6kn days. Finally, we demonstrate that there are temporal realisations of sub cubic planar graphs that cannot be explored faster than in Ω(n log n) days. All these improve best known results in the literature.