With friction shifters, you can readily replace a standard (not "ultra") 6-speed freewheel with a 7-speed, since they are of approximately equal total width. You do not need a 9-speed cogset.
If you replace your 6-speed 14-28 with a 7-speed 14-32 (~$50) and your 26T grannie ring with a 24 (~$15), your bottom gear ratio will be lowered by a very noticeable 20 percent.