I happen to working on my tranny today.. so if this is the spot your talking (photo below) on a TH400, I think the 350 is close to the same location.. But that's a kickdown switch.
1957 Thunderbird 289 1967 Firebird Base 461 1968 C-20 327
That wire is for a lockup solenoid on that TH350. I am told by my transmission guy that those cases are getting harder to find. My dad has one in his 77 Camaro. When he took it in to have it rebuilt, the transmission guy begged him to sell it. It was the first time I had seen one. For that to work correctly, you would need to install a lockup style converter.
Well had a chance to look closer at the wire on my THM 350, I'm pretty sure it is an electric speed sensor of some sort as it comes directly out of the speedo housing..