It depends on the type of transfer and which systems are involved, and the answer is different for each scenario.
For federal to federal transfers, the approved call list is tied to the inmate's register number and travels with them through the Bureau of Prisons system. There is no formal waiting period, but there is typically a practical gap of a few days to a week while the inmate goes through intake at the new facility, gets assigned to a housing unit, and gains access to the phone system in their new unit. The list itself does not need to be rebuilt, but getting physical access to a working phone in the new environment takes time.
On the TRULINCS email system, that is also tied to the federal register number and should remain accessible once the inmate is set up at the new facility, though there may be a brief gap during the transition.
For transfers between different systems, such as county jail to state prison, county to federal, or state to federal, the gap is longer and more significant. These are completely separate systems with no shared infrastructure. Phone accounts, commissary accounts, and contact lists do not transfer between them. Everything has to be established fresh at the new facility. That process typically takes about a week from arrival before phone access is restored, sometimes a few days faster at facilities with efficient intake operations.
During any transfer gap, physical mail is the most reliable way to stay connected. Letters sent through USPS or InmateAid continue to reach inmates even when phone access is temporarily unavailable, and having something arrive in the mailbox during that disorienting first week at a new facility matters more than most people realize.