Should I go for mechanical engineering then aero, or go aerospace all the way?

I am trying to figure out whether it would be smarter to get a BS in Aerospace or Mechanical engineering, and then go on to a Masters in aerospace. Which would you suggest?

I am entering my senior year of high school and am really passionate about one day working for either SpaceX/NASA or the like. I have not taken a jumpstart on engineering or tech classes since I have only recently became sure of my desired major, but have long been a lover of space. Now that it is time for me to start applying to schools I have to be sure what exactly my end goal is. Is it smarter for me to earn a broader degree such as mechanical engineering so that my job options will be more open, even though I eventually want to specialize in an astronautical degree. Or should i stick with aerospace the whole way through? I hope you see this and thank you :)