Shoe For Peak Bagging

I am mostly a trail hiker and have been using trail running shoes, specifically Brooks Cascadia 4, for most of my hikes, and backpack trips. Recently I joined a "Peak Bagging" hiking club that does lots of off trail and rock scrambling in the Sierra of Northern California. (Norcal Peak Baggers out of Sacramento.) Do I need to get higher shoes or boots for that kind of stuff?