Originally Posted by
Al1943
When you attach the cable to the derailleur the shifter needs to be in its smallest cog position with all of the cable out of the shifter. Flip the smaller shift lever several times while pulling the slack out of the cable.
Low normal derailers have a reversed set up so they default to the biggest cog and pulling the cable brings the chain down to the smallest cog.
Thinking that in this case that, if the cable is new and was not damaged the housing needs to be looked at... I really suspect that a worn rear housing is the most likely culprit.
You can check this by seeing of the cable pulls through this section of cable smoothly.
Woould also like to see a side shot of the rear to make sure the rear housing is not too short and causing issues.