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While you don't have to buy the tail coverage from the same company, it might be easier since they already have all your information. If you can find a better rate though, it's fine to go with another company.
I know the tail coverage covers events that occur while the policy is in force but reported after the E & O policy terminates. What about policies that I have written that have a claim after the termination date.
Tail should be written to cover them. It's something you need to discuss with whoever is going to carry your tail coverage.