Adding gears to a single speed Cycle
If you're willing to accept 5 gears, it's really not that difficult. The main issue is finding a hub that's narrow enough to fit the dropouts.

Sturmey Archer make a wide-range 5-speed geared hub, with or without coaster brake, with 120mm OLD spacing. There used to be a 5-speed SRAM hub with 122mm (?) OLD, but it might be a job to track one down. If it's a steel frame with thin stays, you have a couple more mm of leeway. But most hub gears are 135mm OLD. Depending on the hub, it may be possible to remove axle spacers, but don't bet on it... anti-rotation washers don't count; they mustn't be removed!

Unless your current frame is a conversion, you may have a few cable-routing headaches too (although clamp-on cable guides aren't difficult to find).

It only really makes sense if you love the frame. Part exchange would probably be your best bet, but if you have a good reason to keep the bike then I say go for it.

My partner is very short (5'), and standover height is an issue, so she rides a kids' Fuji track. The cost of the Fuji, plus that of building a new wheel around the SA hub, was significantly lower than any WSD we could find.

Afterthought: A cheap and good way (albeit one that requires some tinkering, swearing, and an awful lot of luck):

5-speeds from the 50s tended to be 120mm OLD, so you might be able to swap out the SS freewheel for an old-fashioned thread-on 5-speed model. There are quite a few caveats, however. For starters you'll need a fairly narrow hub shell. Then you'll need enough free threads on the axle to shuffle the cones and locknuts off to one side, so there's enough axle showing to fit the wheel securely. Then you'll need to re-dish the wheel.

Then you'll just need a rear mech with a bolt-on hanger claw (assuming the frame doesn't have a threaded hanger on the dropout). If you're friendly with a local bike shop / junkyard, you might be able to blag these bits and pieces for free.