Starting issue - starts sometimes - other times just a single loud click

[gotmud]

I need help bad!
1968 C10, 307, 3 on the tree, no a/c.

From time to time (more often than not) when I go to start my C10, the starter makes a single click noise and the red brake light comes on. If I continue to try and start it....each time I hear a single click noise from the starter.....it will start....sometimes on the 3rd try....sometimes on the 15th try.

I replaced the battery today. When the truck is running, a multi meter at the battery registers 14.5 so I assume it is charging. I took the starter to Auto Zone and they tell me the starter is good.

Any ideas? I'm getting ready to head out on a road trip and this has me pretty worried.

Thanks in advance for any words of wisdom!