Is It Possible To Install Another Switch To Control Fan?
It looks like you have "Romex" (NM) wiring now, so you don't have the option, at least not easily, to run another wire to go to the fan.  If you have normal metal conduit, you could use a "fish line" to pull another wire.  If that were the case, you could swap out the current switch with a "double" switch without changing the box. 

But, this can sometimes be tricky in any case, since you don't always have a straight shot from your switch to your fan (which may well have been wired intended to only be a light fixture).

The 2nd white wire coming out is a mystery to me.  White wires are typically the "neutral", so having it come in to this box and then go back out (with the wirenut to connect) makes sense.  But it's odd that there would be a second one in the current cable coming in -- if the cable did have 3 wires plus ground, they would normally be, for examample, white, black, and red.