Before you start hosing out the shifter think about what is happening. The cable is going slack ... that means the shifter is feeding it out (like it should) and the derailleur isn't taking up the slack. Since it shifts okay up to the larger cogs obviously it's not a low normal derailleur because given the installation procedure described it would only try to shift to higher gears and never move from the little cog. If the derailleur wasn't pulling down to the little cog you'd have noticed that when the cable was disconnected and since that wasn't a problem before you removed the cable it's unlikely to be the problem.

As operator said, it's most likely to be the cable snagging on something. Basically anywhere the cable is slack (or beyond all the way back to the shifter) isn't the problem. Look for where the cable is still taut and see what is binding it. If you have no tension at the clamp point of the derailleur then the derailleur might be the problem, but as pointed out before that's less likely. The quickest approach I can think of would be simply undo the cable clamp and (1) verify that the derailleur now pushes the chain back to the little cog and (2) feed the cable back and forth by hand and see if it appears to be sticking anywhere. Likely culprits will be either the inner cable has a kink or flat spot on it or the housing has a bur on the end where it was cut which is dragging on the cable.