My guess is that the shifter is in about 7th gear while the derailleur is on the smallest cog. Then the shifter can only move 3 steps to 8th, 9th, 10th gear.

Disconnect the derailleur and let it naturally find its "resting" position on the smallest cog. You may need to move the chain down with your fingers to do this.
Then Pull the cable by hand and shift the shift lever so the cable comes as far out as it will go.
Hook the derailleur up again to the cable, with the cable snug, but not too tight.
It should shift through the whole range of gears now.
If not, did you play with the limiting screws on the back of the derailleur? Sheldon Brown will explain how to adjust these.