Indeed you might get a better deal, but remember if you do that you have a gap in cover. It might not be much, but where would you be if a liability claim was made against you in the future for the gap, or if you had subsidence later, there would also be problems. Better to renew if you can with current broker/insurer adn if you want to shop around, do well in advance of next renewal.
Check with the broker to see if they have any credit period with the insurer that might allow renewal to be paid at a date later than the precise renewal date. If so, no problem as the broker will pay the insurers and you pay the broker. This may well not apply to all brokers especially the smaller ones. Speak to the broker.