The tradeoffs worth weighing
Overlap costs money; no overlap costs certainty
Paying two providers for a short time is an expense. Not paying for it means the new provider learns your setup with nobody to ask. For a simple setup a short overlap is usually enough; for an inherited or undocumented one, it is the cheapest insurance you can buy.
Ask for cooperation, or secure access first
Asking the current provider to help with the handover keeps the relationship friendly and usually works. Securing your own admin access first is not hostile; it is simply what every business should already have. Doing both, in that order, avoids having to choose.
Switch now, or fix the setup first
Sometimes the real complaint is not the provider but a setup nobody can explain. Switching providers will not fix that by itself. A new provider who starts by documenting and tidying what exists gives you a cleaner base, but it is extra work at the start, and it should be scoped and priced as such.
Contract timing
Read the notice period and any minimum term in your current agreement before you set dates. Switching at the end of a term avoids paying twice; switching sooner can still make sense when access or security is the problem.