Possibly, there are a number of options Michelin, Continental, Pirelli, Vredestein etc. and there will be a range of different opinions.
For wet grip, who knows... there are plenty of tyre tests on the net.
One thing I will say is this: I do not like running different makes of tyre front to rear - this can have a dramatic effect on the handling balance of the car and braking stability...
So in your situation I would either replace the Bridgestones with the same or replace all four. Given that it is now summer (
) then I would probably stick with Bridgestones on the front as they work pretty well in the warm and look to change all four next time...