신랑(新郞) means "bridegroom," "groom," or "newlywed husband," so it is redundant to add 새 (new) to the word. It becomes even more obvious if you consider the Chinese characters for 신랑:
According to the Chinese characters, 신랑 already means "new husband."
Here are a couple of "by the ways:"
First, 신부(新婦) means "new wife," so it would be equally redundant to add 새 to it.
Second, traditionally in Korea, a couple gets married in the home or village of the wife's parents, where they spend three days. After three days they travel to the home of the husband's parents. This trip is called 신행(新行), which means "new journey." On the journey the husband rides on a horse, and the wife rides in palanquin, called a 꽃가마 (flower palanquin). Here is a link to a drawing that depicts a "new journey."