+1
The original poster states that the cable is slack and the high limit
screw is OK. I assume he means that the bare inner cable is slack on the
downtube between the two cable stops. If so, there is something
preventing the derailleur from moving outward and that something is
between the slack cable on the downtube and the derailleur.
Next follow that bare cable to the chainstay. If it not slack on the
chainstay but slack on the downtube, then there is friction where the
bare cable passes under the bottom bracket. That should be easy to fix
because it's out in the open for anyone to see.
If bare cable is slack at the chainstaty and the downtube, then the
problem is after the rear cable stop: the covered cable and/or the
derailleur. Pull the cable from the derailleur and actuate the derailleur
by hand. Push on the derailleur body to shift to a lower gear and then
slowly release the pressure on the derailleur body.
If the derailleur does not shift crisply into the smallest cog, then the
problem is in the derailleur body. It could be either the cage and chain
do not have adequate clearance at a cog (B-screw, unlikely), the
parallelogram pivots need lubrication or there's something interfering
with the spring (trapped road debris).
If the derailleur does shift crisply into the smallest cog, then the
problem is friction within in the covered cable leading to the
derailleur. That friction could be debris within the cable, frayed inner
cable (unlikely because OP replaced the inner cable), internal frayed
cable in the compressionless outer sheath, sharp kink in the outer sheath
(look at the end caps) or misaligned end caps (look for signs that inner
cable is not centered in the end cap but rubs against it - more likely at
derailleur side).
Good luck.