the contact and spring goes through the steering wheel and makes contact with the top of the steering column bushing. Is this the piece you replaced?
http://www.americanclassic.com/images/SMBC6.jpg

Once the steering wheel is on, you install the contact plate http://www.americanclassic.com/images/HCP5.jpg The horn button "snaps" into this piece. If this is all rusted or corroded inside, it will not make good connection and will not activate your horn relay.

Have you verified your wire that goes down through the steering column is not broken?

Mike
1955 Chevy 6400 ex-flatbed (no bed now!) sold September 2023
In the Stovebolt Gallery
1958 Chevy 6400 flatbed W/dump
In the Stovebolt Gallery
1959 Chevy Suburban Owned for almost 20 years, Daily Driver -- sold May 2016
In the Stovebolt Gallery

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