Chhabra said she was not able to get satisfactory answers about cost trends from her previous PBM.

"I wanted to look at Humira, and I wanted to see what the acquisition cost was," she said. "And then I wanted to understand what Tyson was paying for that; it was very difficult to get to those numbers. Part of this was to really get a partner who can help us organize the information, make sure we understand how to manage specialty, and really looking at how to get the best net cost."

Rita Lebedeva, vice president of marketing for Rightway, cited several key factors behind the decision in an email to BenefitsPRO:

• Significant cost reductions "Tyson Foods needed to address rising pharmacy spending while avoiding rebate games and opaque pricing models."
• Member-centric support."The unique 'pharmacist in the family' approach offers unparalleled support, assisting members in medication access and adherence."
• Full transparency and auditable reporting."Tyson Foods has full access to reporting to make informed decisions regarding their pharmacy benefit."
• Financially aligned partnership."Rightway's admin fee model gave Tyson Foods confidence that every prescription is driven to a true lowest net cost."
• Pursuit of innovation."Tyson Foods continues its tradition of embracing cutting-edge solutions across its operations."

A CVS spokesman told CNBC that although the company no longer will handle Tyson's overall PBM contract, it will continue to provide specialty drug pharmacy services in conjunction with Rightway.

Related: The PBM unbundling has begun: How Blue Shield of California and others are embracing new models

"Historically, we have provided Tyson Foods with significant transparency, including point-of-sale rebates for its members, a custom retail pharmacy network and unique utilization management strategies that resulted in a flat trend over the last several years," company spokesman Phil Blando said. "Our most recent comprehensive bid would have exceeded the 15% savings rate claimed by a competitor and reported by a news outlet."

Most large employers currently work with the three biggest PBMs — Caremark, Cigna's Evernorth and UnitedHealth Group's OptumRx. By the end of 2022, these big three PBMs controlled nearly 80% of the U.S. market, according to a Health Industries Research Center report.

Karen Van Nuys, an economist at the University of Southern California, said that if larger employers turn to alternative PBM, it could improve competition and bring costs down. "If they're presented with a broader variety of transparent options where they can actually kind of see and compare. across different PBM providers what it's going to cost them, I think that enables all of them to make better decisions about which provider to use," she said.