Not quite sure what your overall goal here is to start with.
I understand that the existing switch now operates the over head lights. You want to add a switch for the fan. So I assume the added switch is for the fan; but what operates the fan now? How does the fan now get its power? How do you turn on and shut off the fan now?
Even if the existing switch is a switch loop there is still power there to tap off of. Regular single pole switches do not require a neutral. You simple take the live wire (be sure the circuit breaker/fuse is off first) and remove it from the switch. Remove the existing switch. Now take the new double switch. Attach the live wire to the hot lead of the switch (follow instructions provided by manufacturer). Then attach the existing wire for the overhead light to the one lead on the switch. You will need to run another romex/cable to where the fan is getting its power. if the fan is presently on another switch in another location you must run that cable from the double switch to that switch location. If the fan has an on board start stop switch the you must by pass that and wire directly to the new double switch. One way or the other you need to run another line to the fan from the double switch.
BUT, before you do that take apart the switch (again shut power off). Take pictures of the inside of the box. Make sure we can see all the wires clearly. It is quite possible that since the fan is so close to the switch that if the power goes to switch first and then to the light the switch box may already have a cable going to the fan and feeding that. If that is the case it is just a matter of rearranging the wires in the switch box to get it to work. The only other concern I would have is that the double switches are very deep. Much more so than a normal single pole switch. So let's see how many conductors you have first, let's figure out what goes where and then we go from there. I am just hoping that there will be enough room in the switch box for the new double switch. Before you remove any wires from the existing switch be sure you make a little diagram of how and what wires go to the switch and what screws of the switch. Take pictures for yourself for later reference.