Program disturb is reduced in a non-volatile storage system during a program operation for a selected word line by initially using a pass voltage with a lower amplitude on word lines which are adjacent to the selected word line. This helps reduce charge trapping at floating gate edges, which can widen threshold voltage distributions with increasing program-erase cycles. When program pulses of higher amplitude are applied to the selected word line, the pass voltage switches to a higher level to provide a sufficient amount of channel boosting. The switch to a higher pass voltage can be triggered by a specified program pulse being applied or by tracking lower state storage elements until they reach a target verify level. The amplitude of the program voltage steps down when the pass voltage steps up, to cancel out capacitive coupling to the selected storage elements from the change in the pass voltage.