Most of the relays used in older radios have plastic covers that can be carefully removed, allowing access to the contact points. Once access is gained use of tools made for burnishing relay contacts can be used to polish the contact surfaces. Do not use highly abrasive sandpaper to clean relay contacts. If you cannot get relay burnishing tools, then use the finest sandpaper sometimes called crocus cloth to polish the contacts. Then replace the cover.
I had a '56 model years ago that had the tube type search tune AM radio and it always worked OK. I never did have to dig into the radio while I owned the car. There is the possibility that the drive circuit for the relay could also be having trouble. The drive motors for the search tune feature would likely be difficult to replace since they have not been made for many years. There might be some still available through vendors who are listed in the service section on the club website.
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