More voters around Austin, Texas, turned out for the midterms than for the last presidential election, a rarity in politics. Fifteen counties in Montana produced more votes in last week’s election than in 2016. In Georgia, vote totals were near or above presidential-year levels across most of the state.

Far fewer voters participate in midterms than in presidential elections. But interest in this year’s races was so high that vote tallies in some places hit or exceeded presidential levels.

Nationally,...