Changing a Child's Name: Last Name Change of Toddler

My question involves name change laws in the State of: New Jersey

After my child’s 1st birthday, my husband and I legally changed their birth certificate to have their last name hyphenated, legally giving him/her mine and my husband’s last names. This was changed/updated from their original birth certificate with only stated my husband’s last name.

Now that I have a correct and legally updated birth certificate, what documents do I need to update their social security card?

Thank you!

Changing a Child's Name: Last Name Change of Toddler