Following revelations that the man charged with murdering Iowa college student Mollie Tibbetts is in the U.S. illegally, immigration hawks have cried for more stringent employment laws.

But the confusion over the status of Cristhian Bahena Rivera, the 24-year-old suspect in the case, highlights the difficulty in navigating U.S. immigration laws -- even for businesses that fully attempt to do so.

Rivera's now-former employer, Yarrabee Farms, on Tuesday said he had passed an employment check. "This individual has worked at our farms for four years, was vetted through the government's E-Verify system, and was an employee in good standing," Yarrabee Farms said in a statement Tuesday.

Get Breaking News Delivered to Your Inbox

During an appearance in court on Wednesday, Allan Richards, an attorney for Rivera, seemed to allude to Yarrabee Farms' initial statement to argue that Rivera "has legal status." In a court motion, he told the court that Rivera "has complied with his documented status since arriving in the U.S.A. as a minor."

However, the farm's owners said in a press conference later on Wednesday that the company had determined their ex-employee had used a false name to pass verification. They also explained that while they thought they were using E-Verify, they were actually using a different older system through the Social Security Administration.

"Just within the last four hours we have come to learn that the Social Security Administration employment verification service is not the same as E-Verify," said Dane Lang, co-owner and manager of the farm. "We're signing up for e-verification, and we intend to fully vet everyone through the E-Verify system."

Iowa is not one of four states that requires employers to use E-Verify for all workers.

Rivera didn't request or receive any DACA status, which could have permitted him to legally work in the U.S., Citizenship and Immigration Services said in a statement. "We have found no record in our systems indicating he has any lawful immigration status," CIS said.

But whatever Rivera's status, it's far from certain that using E-Verify would have revealed it. Despite calling itself "currently the best means available to electronically confirm employment eligibility," E-Verify doesn't actually prevent companies from hiring undocumented workers, according to studies of the system's efficacy.

In a report for the Department of Homeland Security, research company Westat found that while so-called authorized workers have an error rate of just 1 percent, unauthorized workers get a pass at a rate of 54 percent.

"The way to view it is, if you know a worker is an illegal immigrant, E-Verify is worse than a coin toss," said Alex Nowrasteh, a senior immigration policy analyst at the libertarian Cato Institute.

According to the report, "The substantial estimated inaccuracy rate for unauthorized workers is not surprising," because many of these worker submit documents that are in fact genuine, but are not their own, the report said.

"The big loophole is the system checks the documents I give you. It does not check you," said Nowrasteh.

As of April, about 800,000 employers were signed up for E-Verify, per CIS. That's a small fraction of the more than 3.6 million U.S. businesses with two or more employees, according to the Census Bureau.

CBS News' Graham Kates contributed to this report.