The reverse lights never came on when the car was in reverse in my 240Z. Twenty-four years ago, I replaced the switch, but the lights still didn't come on.

I decided to try again. I installed a new OEM switch and checked the leads for continuity. No go. I moved the gear shift and found that the transmission pressed on the reverse switch in 4th gear.

I assume there's no way to fix this without opening the transmission. 

I forgot to mention that the transmission is an early 5 speed from a 280Z. 

No other ports for a switch?  The 4 speeds had a top gear switch, my 76 had one.  It activated the vacuum advance solenoid.  The ZX 5 speeds had several switches.  Looks like some had top gear and OD switches, for some odd reason.  Maybe they did the same thing.

For some reason Nissan didn't label the switches in the 280Z chapters.  But I think that they'd be about the same as the ZX's.

Thanks @Zed Head. I moved the switch to the port closest to the clutch fork. The switch makes contact in reverse. 

As I looked more carefully at the transmission, I realized I have the truck 5 speed.