The short answer is, of course, no. At least not at the same time.
Typically, travel nurse staffing agencies write clauses into their contracts that prohibit you from returning to a particular facility under a contract with a different agency. Some agencies will ‘allow’ it but only after the original agency is paid a ‘nominal’ fee of about $8,000.
Also, many healthcare facilities won’t allow the practice, either. It’s possible that they could take on a travel nurse as a permanent staff member after their contract has expired and pay the contract fee to the agency, but it’s very rare that a travel nurse change agencies and remain in the same facility .