Do read the policy and/or ask whoever it is you take the new one out with, but I would expect it broadly to be as follows.;
If you take out a new policy with a new company and something occurs after you've taken the new policy out and the old one has expired, then the new company would be the one to claim from. If you need to cancel the holiday before the new policy is taken out then claim off the old one.
So, If you go to the doctor next week and are told you need an operation that means you can't travel in February,then claim on your old (existing) policy.
However if in Febraury you're on your way to the airport, break your leg and can't travel then claim of your new one.
What matters isn't when the holiday was booked but when the event happened that caused the cancellation.