Apparently there's a good chance some seal will fail from being dry after so long on the trans if you just fire it right up. I would drop it same time you're working on the engine and do a rebuild kit. Others I'm sure can give more precise advice.
kanter.com/p-35861-ultramatic-rebuilding-kit-1951-1954-transmission.html Watch Ross's videos on trans disassembly here, should be pretty similar to yours
youtube.com/@speedwellgarage8588 What's nice is you have a convertible and they hold value pretty well, spending a few grand on parts to get it going good is a no brainer. On the HT/sedans there is more of a monetary concern.
As I hear those engines are quite nice to work on, even after 40 years I think you'll be able to get it going again if you're careful about things as I'm sure you will be.
As others have said, pics/updates are greatly appreciated!

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