Some banks offer sweep accounts that automatically move funds over the 250k limit into accounts at other banks (again, up to the 250k limit) or into money-market funds (which have their own risks).
I believe there is no way to eliminate credit, default, and/or liquidity risk entirely in banking (when cracks in the system appear and start spreading, you may have exposure no matter what you did), but it's possible to mitigate it in ways that don't seem to have happened here.
Unless they use sweep accounts, accepting that those accounts can in some cases have different pros and cons per bank such as early withdrawal fees and minimum limits to avoid fees. Sweep accounts can raise the FDIC insured amount per depositor.
I believe there is no way to eliminate credit, default, and/or liquidity risk entirely in banking (when cracks in the system appear and start spreading, you may have exposure no matter what you did), but it's possible to mitigate it in ways that don't seem to have happened here.
Here's a decent explainer on sweep programs (it talks about broker-dealers using them, but banks can also offer them): https://www.sec.gov/oiea/investor-alerts-bulletins/ib_banksw...