There's always going to be a some water, (should not be a great deal) in the bilge after it pumps out on stand-by. The float switch can only go so low. Even running the pump on manual, there will be a little, it depends on how close to the transom the pump is.
What type of set up do you have. An auto pump, with self contained float, or a pump and seperate float switch? If it has a seperate float switch, you should be able to manually move the float arm to check it out. Poor some water into your bilge and see what level the pump comes on. The only way to adjust it, is by physically lowering, or repositioning the float switch. Sometimes, you can reverse the position of the switch, (arm facing the stern) and the pump will come on sooner. If you do move it, make sure you use sealent on the screws.
Also, usually the the float switch will operate the pump in either off or standby, if it is directly connected to the battery. If it only works in stand by, that means it is connected to the boats main battery switch, (isolator). Nothing really wrong with that, but if you slip the boat, you cannot turn off the batteries, because the bilge pump will not work.