I recently watched an FB Live with Tim Holtz this week and he spoke about this issue. (Check out his FB page for Tim Holtz Addicts to find the video.) A tip he shared was to take one of cutting pads from the larger Sizzix or Vagabond die cutting machines and secure the sidekick to that. Then when cutting out the die you can press down on the cutting pad the sidekick is secured to while turning the sidekick handle to cut the die. Hope this helps!
Of course if you don't own a larger cutting pad you'd have to find a different solution. Perhaps a large 12x12 smooth tile? Sometimes one might have something like that laying around. (I've done a lot of different crafting so I have lots of odds and ends. Most crafters, er hoarders, do. LOL)