I've replaced the 5-speed freewheel with 6-speed standard-spaced freewheel quite a few times where no axle respacing was needed.

Having the smallest cog closer to the dropout is ideal unless it causes a chain contact issue, better chainline and the strongest axle and wheel geometry when you can use fewer driveside axle spacers.
A narrower chain can often make the difference if some contact with the stays were to occur, and usually improves shifting as well. I would usually use 8-speed chain with any Asian 6-speed freewheel, but 9-speed chain often seems even better.