Birkhead is seeking custody of 6-month-old Dannielynn and filed paternity and custody suits in both the United States and the Bahamas, where the child was born and currently lives.
Also claiming paternity is Howard K. Stern, Smith's companion at the time of her death Feb. 8. He is listed as the father on the child's birth certificate. Smith's estranged mother, Virgie Arthur, also seeking custody of Smith's daughter, attended the hearing in Nassau Friday.
"You just hope for the best and I'm pleased with what I heard today and I'll stay as long as I have to stay to get done what I need to get done," Birkhead said.
A DNA expert is expected to testify when the hearing resumes next week, "Access Hollywood" said.