Could still be all of the above.
The sound you describe is still the starter relay/solenoid on the fender making it's clicking sound, but with a little more power available to it.
A really dead battery, or other defect in the starting system will often give you that one loud click. But a going-dead battery will give you the torque-brake sound of a drill-driver reaching it's torque setting.
But other things that can cause that sound are:
1. bad starter relay.
2. bad starter
3. stuck starter (see bad starter)
4. bad connections between the key and the relay.
5. bad connections between the relay and the battery
6. bad connections between the relay and the starter.
A new alternator is NEVER a guarantee of working anymore.
For one thing, it could be connected incorrectly. For another the regulator might be going bad (did you replace that at the same time?). For a third, the battery and starter cables might just have chosen this moment to coincidentally go south for the winter. And unfortunately too, new parts are no guarantee of working. Did you verify 14.5 volts with the engine running after you replaced the alternator? With the battery maybe going dead immediately after a new alternator install, that would be everyone's first thing to suspect. Why did you replace it in the first place? And did you check the battery voltage at some point after your trouble started? If not, you should.
And lastly, even a reasonably good battery can go bad without notice. More often in very hot or very cold weather.
So you still have some things to check.
Paul
