In online graph exploration, an agent must visit every vertex of an unknown weighted graph and return to its start, learning the incident edges at each vertex it visits. On $n$-vertex graphs, the best known competitive ratio has been $O(\log n)$, achieved by several algorithms, among them the hierarchical depth-first search of Megow, Mehlhorn, and Schweitzer. We give an algorithm with competitive ratio ${(2+o(1))}\ln n/\ln\ln n$ for known~$n$. Like hierarchical depth-first search, it groups the edges into geometric weight classes and searches depth-first between components of lighter edges. Unlike it, the agent tours each component along a minimum spanning tree for the original weights, so only crossings between components pay for rounding, and the base $b$ of the weight class can grow with~$n$. The leading constant $2$ is tight for our choice of base. Without knowing $n$, restarts with increasing size guesses give upper bounds with leading constant $4$ for a fixed schedule and $2$ in expectation for a randomly shifted schedule. If the weights lie in a known range $[1,2^k]$ with $k\ge4$, choosing $b=k/\log_2 k$ gives a competitive ratio of $O(k/\log k)$ without knowing $n$.

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