못내 means "immeasurably; incalculably; deeply" ex) 그녀는 그와의 이별을 못내 아쉬워했다 She felt deep regret about having to part with him. 할머니는 내가 가는 걸 보시며 못내 섭섭해하셨다 My grandmother was grieved to see me go. 못내 한번 더 참았던 그 말 means "The word that I had kept deeply once more"

It emphasizes 참았던 and suggests some lingering feelings.