There are two means you can use to adjust. Since the rear derailleur seems to reach the large cog in the rear, I don't think adjusting the cable is needed.

The H L stamp on your Shimano derailleur is the indicator which screw will adjust the outboard and inboard travel of the derailleur. The rear derailleur will have something similar, though it may not be marked. It sounds as though you need to modify the amount of inboard travel (will be "L" or "Low" if it is marked). Ideally, when the derailleur is at the limit of it's inboard travel, the small pulley wheels will be directly aligned with the largest cog. If it isn't, turning the "L" screw counterclockwise will allow more inboard travel, and clockwise to allow less. Adjust 1/4 turn at a try.