The measure would increase the wait time from one hour to one day. It had tied the clock to an ultrasound, but a compromise approved on Wednesday in the Senate removed that requirement.
Women who choose to get an ultrasound the day before can get one at crisis pregnancy centers statewide.
Abortion rights supporters applauded the removal of the two-trip requirement but said the measure still requires medically unnecessary delays.
Supporters hope the 24-hour wait will lead to fewer abortions.
The measure had stalled in the Senate for a year. The tweaked bill will return to the House.