Figure out the required capacity of the mech (big ring + big sprocket) - (little ring + little sprocket). See if your mech can handle it: The techdoc for your mech will tell you the capacity of your mech and also the largest sprocket it can deal with.
Rear mechs are surprisingly cheap, so swapping a GS for an SS may not cost much.
But beware ... you could end up with quite a shopping list. New mech, new cassette, new chain, new gear cable, oh ... and what state are your chainrings in? And while you're at it why not replace the other cables too and ... and that bar tape looks a bit tatty ... and ... and.