Larry Birkhead, the baby's father and Smith's former boyfriend, said he hoped to be leaving the Bahamas "soon," though the court set another hearing in the case for Wednesday.
"On my behalf, everything went well," Birkhead told reporters after the hour-long hearing. "I'm moving toward starting a life with my daughter outside of the Bahamas."
Birkhead and Arthur have tried to reach a visitation agreement that would end her legal attempt for custody of Dannielynn, who could inherit a fortune from the estate of her mother. Smith, 39, collapsed and died from an accidental drug overdose in February.
A DNA expert revealed earlier this month that Birkhead was Dannielynn's father. Smith's companion, Howard K. Stern, has said he supports the former boyfriend's custody bid.
Stern, who also attended Friday's court hearing, said the case "should have been over today," before he left with Birkhead in a black Cadillac sport utility vehicle with tinted windows.
Smith gave birth to Dannielynn in September in a Bahamas hospital, days before her 20-year-old son, Daniel, died from a lethal combination of drugs at her bedside.
The baby has been living in an oceanfront home with Stern, who was listed on the birth certificate as her father, but Birkhead has been visiting her in recent days to get acquainted with her.
By Jessica Robertson