When you changed the high E string did you use the same gauge strings that were already on it? (Probably .009 was what was on it to start with.) If you used a higher gauge (thicker) string (e.g. .010 or .011) then when you bring that string up to pitch, it's going to be under higher tension, and raise the bridge up, and lower the tension on the other strings, so when you bring those strings up to pitch, the bridge is going to come up... etc.
So, I'm guessing the new E string is a thicker gauge than what you started with... otherwise, changing just one string shouldn't be such a huge effect on the other strings, even noting that the floating bridge is of course a pain in the ass when it comes to tuning compared to a fixed bridge.
You may have to make some adjustments as noted by others to compensate for this, or if I guessed right, and you used a thicker E string, maybe change to an E string from a set that's the same gauge as what you started with and leave your setup alone... Just an idea.