After Thursday night’s Democratic presidential debate in Los Angeles, Amy Klobuchar slept for about 3½ hours, appeared on cable television shows before sunrise and then flew to Iowa, where her green campaign bus was waiting.

Traveling along the Iowa-Missouri border, the Minnesota senator made stops at coffeehouses and restaurants late into Friday evening and then appeared at 10 events on Saturday, all in an effort to lock down votes ahead of the first-in-the-nation Democratic presidential contest in February.

“I...