You definitely can! Infact this is a good idea. There are also some screwdrivers that also have a retaining clip. I have seen them a few times. I infact would like to find one. I myself use a magnetic screwdriver for many types of mounting applications. I have a big button magnet that I keep for magnetizing screwdrivers. A few swipes does the trick, if the magnetism goes weak. I keep magnetized screwdrivers away from delicate things such as mechanical meter movements, which are getting rare these days, and only found on very specialized equipment.
Infact you can say that the screwdriver has a magnetic personality!