Some said they had paid traffickers a relative fortune of $1,000 for a place on a boat.
Bangladesh is home to one million mostly Muslim Rohingya refugees, the majority of whom fled a violent 2017 crackdown by the Myanmar military that is now subject to a United Nations genocide probe.
Conditions in the overcrowded, dangerous and under-resourced relief camps are tough, and refugees have said conditions are worsening due to cuts in food aid, deadly gang battles and a lack of jobs.
As the seas in the Bay of Bengal calm after monsoon rains, human traffickers are offering hundreds of Rohingya people berths on boats bound for Malaysia and Indonesia, police officers told AFP.
This month has seen a spike in journeys to Indonesia's westernmost province of Aceh -- a voyage of about 1,800 kilometres (1,120 miles) -- with more than 1,000 arrivals in the biggest such wave since the 2017 crackdown.
Police said they had "stopped and held 58 Rohingya" on Friday night as they reportedly left camps heading to board boats at Teknaf, a Bangladeshi river port just across the border from Myanmar.
"Among them nine are men, 16 are women and 33 are children. We detained two Bangladeshi human traffickers who were allegedly guiding them," Teknaf police station chief Osman Goni told AFP on Saturday.
"They were ready to sail in a boat from Teknaf and were going to Indonesia and Malaysia."
Several Rohingya refugees who were held by police told AFP that they had paid human traffickers around $1,000 each for a boat journey to Indonesia.
"They (the traffickers) kept us in a house, and said they would send us to Indonesia on a boat the next day," said Selim Ullah, a 22-year-old Rohingya refugee.
Ullah's wife and daughter were also held by police.
© Agence France-Presse