We consider submodular maximization under increasing cardinality constraint and ask for a good incremental solution, i.e., an ordering of the ground set such that each prefix of the ordering yields a good solution for its respective cardinality. A classical result in this setting is that the greedy algorithm achieves a competitive ratio, i.e., an approximation guarantee across all cardinalities, of $\mathrm{e}/(\mathrm{e}-1) \approx 1.582$. No better general guarantee was previously known. We present an adaptive scaling algorithm achieving a competitive ratio of $1.373$. We complement our result by a deterministic lower bound of $1.25$ on the best possible competitive ratio for incremental submodular maximization.

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