KANSAS CITY, Kan. (AP) — When Planned Parenthood decided four years ago to open a new clinic in a medically underserved working-class neighborhood here, it envisioned a place that would save women living nearby from having to take hourslong bus rides to obtain birth control, testing or an abortion.

The U.S. Supreme Court's June decision overturning Roe v. Wade — four days before the clinic opened — changed all that. Because Kansas is one of the few states in the region where abortion remains legal, the clinic soon found itself inundated with calls not just from panicked patients in Kansas and...