Starter mortor or solenoid BAD?

I have a new battery in my car. 3.0L I6 engine. I live in north Dakota and during winter it goes below zero most of the time. i don't have a garage and when it extremely cold out side (-40 with wind chill) my car would still but right after it starts i hear this click with a short decreasing buzz sound. ( sounds like a clamped thin metal strip being pushed down and released) this didn't happen happen in summer. Does this mean the solenoid is going bad?