So, the right and left lamps on you car, have only one socket each, and that socket has only one contact?  Possibly, then, a former owner has fooled with  this car.

Normally in the late '30's each rear lamp had one socket with TWO contacts, and the socket held a mating double-filament bulb with two contacts..  One contact (in each socket) powered the dimmer of the two filaments in the bulb (the "taillight" function). .  The other contact in the socket, powered the brighter filament in the bulb. (The "brake light" function.)

The brighter filament serves both as the "brake light" and "turn signal".  So -- if the socket in each lamp of your car has but one contact -- you must purchase a socket with two contacts, and mount it inside the right and left taillight lamp. I think these are fairly easy to find. You will remove the old one-contact socket and somehow install the new two-contact socket in its place.  CAUTION:  these are bayonet-style bulbs so be sure that they will fit into the sockets you buy, so that the contacts are correctly aligned with one another after installation.

Your conversion can be done with a Step-Down turn signal switch of the 6-wire type. It makes for a lot neater installation than a modern, one-size-fits-all turn signal switch.  Be sure to get the switch (with its handle and knob), the metal raceway that carries the wires down the steering column jacket (from the switch), and the cast metal chrome molding that carries the wires up under the dashboard.

You will have to change the wiring under the dash.  The brakelight wires will now go into the signal switch.  When you are driving and activate the right signal, it will blink.  If you put the brakes on, the right lamp will blink, and the left lamp will illuminate steadily.

You can make your Hudson switch self-cancelling by cutting a small hole in the side of the steering column jacket so that the "fingers" from the switch, penetrate the jacket.  Then, attach a small hose clamp around the steering shaft.  When the shaft rotates (with the steering wheel), the screw on the hose clamp will hit the "fingers" on the switch, and will cause the switch to cancel.

You can use your car's front fender lamps as the front turn signals, if you're lucky to have a pair.  Otherwise, you can buy some L.E.D. lights and mount them somewhere up front until you find a pair of fender lamps.

I did this on my 1937 and it works perfectly.  It also looks like it was made for the car, even though the switch is from a '52 Hudson.

However, you must have dual-contact socket in each rear light.