The front derailleur's position when it is in the middle-ring position is determined by the shifter, not the derailleur. But the low limit screw does play a role in determining where the shifter places the front derailleur when moved to the middle ring position.

If the low limit screw is set so that the inside cage of the front derailleur is not close to the inside (low) ring, it won't shift properly to the middle ring. As that limit screw is tightened and the cage moves closer, the front shifting will improve such that it makes the jump to the middle gear. However, if the cable is tight after making the low limit screw adjustment you may need to take some of the tension out so that the front derailleur can swing as far out as it needs to and this is where middle ring performance is most greatly affected.

Return the chain to the low (smallest) front chainring and low (largest) rear sprocket and check the cable tension. If you've already made the low limit adjustment, let out any cable tension by loosening the cable clamp bolt and let the tensioned cable slide back through. Now tighten it up and take out any remaining slack with the cable adjusters. (You may want to dial in your cable adjusters before you start.)

Now your shifter will begin it's swing where you set the low limit screw. With the bike in low front and low back, the inner front derailleur cage will need to be near the low chainring. (It can get pretty close when in this gear combination because shifts to a higher gear will only move the chain further away from the front changer cage.)

If you still have problems with the shift from middle to low chainrings, check first that when the changer is in low that there's little or no tension on the cable. If that checks out, back out the low limit screw 1/8 of a turn and check to see if the middle to low shifting has improved. Repeat as necessary keeping an eye on cable tension and using the cable adjustment barrel to compensate for too much or too little tension.