You could attach the tether at a higher or lower latitude, but standing on the ground next to it, it would appear to rise into the sky at a rather alarming angle (more or less parallel with the equatorial plane). More of the cable would be going through the atmosphere. Centrifugal force would hold it taunt and a great deal of tension at the base would be required to keep the entire system in that configuration. At no point along the cable would the cable be following a valid orbit (the center of mass of the cable would not be in geosynchronous orbit). You would need a much stronger cable, and the result would be far less useful.

Edit: http://gassend.net/publications/NonEquatorialUniformStressSp...