I'm not sure on this one, my first thought would have been to change the inner/outer cables, but you said that it was like this prior to changing both recently - so that would indicate doing something else.
What would be ideal would be to post a picture here square on from the rear with the bike in the big ring front, and say third gear from the left at the back. This would allow us to see better what the derailleur is doing, if you can get the hanger in too, that would be a bonus.
As has already been said above, the mech will either work or it won't - the most common cause is damage - but if you don't keep it cleaned properly then it can start to cause minor issues so it would probably be worth taking the mech off completely, and giving it a proper deep clean - degrease and re-lube all the pivots making sure you've got all the muck out, then same with the springs. Take off both jockey wheels, disassemble them completely and clean them up - I take the bushings out and then pack them with lithium grease - they don't move as freely at first, but I find it helps keep the crap out a bit. Also check that the cage is not bent.
Then put it back up in the workstand and reconnect everything up - take the opportunity to re-index the gears from scratch.
