1965 Dodge Dart GT will start in neutral but only sometimes in park

Hi all,
As noted, the car will always start in neutral but only sometimes in park. This is a floor shift model. My first guess would be the neutral safety switch, but does it only function when in park and not neutral? Thanks for any info...
Tom
I have adjusted the cable to where it should be consistent...but the steel "cage" that supports the shift/cable mechanism and mounts the shifter itself has been broken and rewelded by the previous owner so, that may be a factor. I've spent many hours getting the setup to be good with regard to the shifting position and have replaced cables, etc... so the tranny (rebuilt) shifts fine with relation to the shifter location. Is there a trick to dialing in the exact cable position (at the cable clamp location in the shifter housing) or can the neutral safety switch be adjusted at all...it doesn't look like it can be. It seems like the neutral safety switch should be OK if it works for the neutral position. No?
Tom