"I probably should've mentioned I believe the cable needs adjusting because the temperature slide control only moves halfway into the blue and stops."

Ha! That could be why there's not much cold air. The cable is a simple one and you can "move" the end metal sleeve (silver barrel) forward or backward to adjust the degree of control, then retighten the sleeve holder screw (...for lack of proper term). If yours is the "earlier" type heater valve you can adjust the end length of its inner cable at the valve lever (adjustment screw is there at lever).



"The small screw on top is too close to other components of the car to get a small screwdriver in there."

Maybe you could undo the screw via sideways. Use a pair of pliers to grip the screw's sides and twist it anti-clockwise (if viewed from the screw's top).




But first, undo the cable at its heater valve end. With the cable undone, play with the valve lever to ensure its able to close properly. The heater valve could be stuck open too (rusts inside?) and that could be why you couldn't fully push the temp lever to extreme left (blue). If the valve is stuck then you'll have to replace it (and this includes first draining your radiator coolant to prevent flood in your cabin..installing new valve...refilling it later..and so forth).

For more convincing troubleshooting (if left up to me), drain your coolant, dismantle the heater valve out and play with the valve lever. You could blow into the valve with lever at extreme positions to see if its really able to stay close or open.

Here's a website for guide:
http://europartshouse.com/index.php?main_page=page&id=7

Regards,
Amarin.