Toyota Repair: 01 tundra 4.7 starter electrical

Question
Voltage issues with cold temp start. is there a a cutoff voltage in the starting circuit that would prevent something other than the start relay which I have replaced? I can hear the relay click in the fuse box but apparently it is being interrupted on its path to the starter. I have a wall Mart battery with known issues...but it will hold a 12.80 volt charge and registers 14volts while the engine is running. If you think it is the starter your opinion would be appreciated I know where it lives and not looking forward to replacing it. I just finished replacing the radiator and tilt box/shifter.so I can do this myself.would the shift relay...neutral safety be affected by a low volt condition? thanks.and btw thanks for helping others. many of your fixes I have done through trial and error.wish I would of found this site sooner

Answer
This is a very common thing on the early and some later Tundras, yes, the problem is the starter, the only other thing that is between the relay and the starter is the neutral start switch, if there is no clicking noise at all from the starter it means current is not getting to the solenoid and the NS switch is the most likely cause, it's located on the transmission on the side where the shift selector shaft exits the housing.