Sounds like what you need is to shift the gearing range up, rather than increasing the number of gears. What are the tooth counts of the upper and lower gears on the back, and the front? You could put bigger chainrings on the front, or smaller cogs on the back. The back is probably easier, as should not require changing the dérailleur. The front might need to change it if you did the change there.

EDIT: Oh! and is a freewheel or freehub?
http://www.sheldonbrown.com/free-k7.html