If it is a regular (not Ultra) 6-speed, the rear spacing is 126 mm and a 7-speed freewheel will easily replace the 6-speed with no other changes.

You could respace the rear to 130 mm (or just force a 130 mm hub into the current dropouts, it's not that hard) and go to 8/9 or 10-speeds but that would require an entire new rear wheel.