Originally Posted by
Vintage_Cyclist
1. You need to measure the bicycle's drop-out (AKA over lock nut) width. A 5 speed frame will probably be either 120mm or 126mm at the rear dropouts. If it's the former, you can only fit a 5 speed or a 6 speed with compact spacing, like a Suntour Ultra (no longer made, mainly reliant on eBay to find). If it's the latter, you should be able to go up to 7 speed freewheel, which is still being made by various companies, including Shimano. A work around, that will allow you to get more gears, is to cold set the frame.
Fitting a standard spaced 6-speed or a 7-speed freewheel to a 120 mm frame will require more than cold setting from 120 to 126 mm. The hub's OLD (over locknut dimension) will also have to be increased to 126 mm by fitting a longer axle, adding spacers and redishing (centering) the rim.