Rodriguez is accused of kidnapping 22-year-old Dru Sjodin from the parking lot of a shopping mall in Grand Forks, where she worked. She still hasn't been found.
Rodriguez is a three-time convicted rapist.
His lawyer says Rodriguez agreed that for now, he should remain behind bars for his own safety.
Family members continued Thursday to express confidence that Sjodin will be found alive.
"I just know she's alive and we're going to get her," Sven Sjodin, Dru Sjodin's older brother, said Wednesday evening, after the volunteers searched fields, ditches and outbuildings in Grand Forks County of North Dakota and Polk County of Minnesota.
He thanked "everyone who came out with their four-wheelers and ATVs and just took the time out of their day to help us find our little girl."
A massive search effort failed to turn up any sign of the missing woman. CBS News Early Show Correspondent Jon Frankel reports 1,700 volunteers participated Wednesday. Some volunteers were left in tears Wednesday as their search for the 22-year-old Sjodin came up empty.
Police Chief John Packett says he looks for a break in the case soon. He won't say if there's any specific evidence Sjodin is still alive, but says, "That's what's driving us."
Rodriguez, 50, was arrested Monday in Crookston, Minn., about 30 miles from Grand Forks, where he lives with his mother. He waived extradition to North Dakota on Wednesday.
Rodriguez did not speak during Wednesday's proceedings in Crookston. At the end of the hearing, an unidentified woman rushed toward the front of the courtroom and spoke in Spanish to Rodriguez. "For your mother's sake, tell us what you did with the girl!" she said.
Rodriguez wore a bulletproof vest while he was shuttled between Crookston and Grand Forks, the same area where searchers used all-terrain vehicles and walked hundreds of square miles.
Police have declined to talk about evidence in the case and the criminal complaint has not been made public. Authorities have interviewed Rodriguez, but Grand Forks Police Capt. Mike Kirby said police would not comment "regarding his level of cooperation."
Authorities have received more than 1200 tips.
Wednesday was the third day that volunteers had helped police and the 10th day of searching by Sven Sjodin, who planned to return home to California on Thursday to be with his wife, who is expecting the couple's second child within two weeks.
"My sister is my flesh and blood and she is one of my immediate concerns, but I also have another immediate concern," Sven Sjodin said. "I know that I have a large contingent of friends who will continue to search until they find Dru."
Family members said they were overwhelmed by another strong turnout of volunteers.
Rodriguez had been released from prison in May, after serving a 23-year sentence for an attempted kidnapping and assault of a woman in Crookston in 1980. Authorities said they had probable cause to believe Rodriguez was in the parking lot of the Grand Forks mall where Sjodin worked at the Victoria's Secret store the night she disappeared.
Rodriguez has a history of sexual contact and attempted kidnapping with adult women, including a guilty plea to aggravated rape in 1975. He has used a weapon in at least one assault, according to the Minnesota Department of Corrections Web site.
His past offenses require that Rodriguez be registered as a predatory offender, the department said. The classification is for those people whom authorities believe are at the highest risk of committing another sex crime.
Officials said Rodriguez was considered for civil commitment in 2001, near the end of his last sentence. But a psychologist decided against recommending him for the program, which could have kept him in custody indefinitely. A special board concurred.