While not part of the rosary, there are other Marian prayers that you may wish to recite.

The following is called the Memorare:

Remember, O most gracious Virgin Mary that never was it known that anyone who fled to your protection, implored your help, or sought your intercession was left unaided. Inspired by this confidence, I fly to you Virgin of virgins, my Mother. To you I come. Before you I stand, sinful and sorrowful. O Mother of the Word Incarnate, despise not my petitions but in your mercy, hear and answer me. Amen.

And one of the most truly beautiful prayers of the church is the Magnificat (sometimes called the Canticle of Mary):

My soul proclaims the greatness of the Lord, my spirit rejoices in God my Savior. For He has looked with favor on His lowly servant. From this day all generations will call me blessed: the Almighty has done great things for me, and holy is His name. He has mercy on those who fear Him in every generation. He has shown the strength of His arm, He has scattered the proud in their conceit. He has down the mighty from their thrones, and has lifted up the lowly. He has filled the hungry with good things, and the rich He has sent away empty. He has come to the help of His servant Israel for He has remembered His promise of mercy, the promise He made to our fathers, to Abraham and his children forever. Glory to the Father, and to the Son, and to the Holy Spirit, as it was in the beginning, is now, and will be forever. Amen.