As an update, I dropped By Express Bike Shop. They sold me four spacers for a dollar, in a variety of thicknesses from .8mm to 3mm. Then I checked my 7 speed freewheel, and it also jumps from 24T to 34T. I could have sworn it had a 28T cog on it. So there's really no need to replace my 6 speed now, but if I find a suitable freewheel I can give it a try.