SirMike1983
Riding a '38 Autocycle Deluxe
Thanks for your replies everyone, I really appreciate it.
I was surprised to find that my front hub does not use any jam nuts to lock the cones in place, and it doesn't use a washer against the cone on the non-drive side. If I install the no turn washer on the drive side the rim ends up centered in the forks and the spacing seems right for the fork, so I'll leave it with that setup.
The photo below shows what I'm ending up with for hardware. It's just a repeat of what you all have said above, but I'm including the photo in case anyone searching this subject again in the future will find it helpful.
From the left; Non-Drive side rear, Drive side rear, Non-Drive side front, Drive side front.
Again, this is on a 1954 Schwinn Traveler.
Thanks again everyone!
View attachment 2221916
Looks like you've got what you need.