Hmmm, from your comment about breaking the letters apart I'm guessing they're Memory Box dies? I haven't tried those and have no idea what thickness they'd usually go through but I'm surprised that cardstock plus tape seems to pose a problem (unless you're using unusually thick cardstock, I guess). Did you try adding a shim?
If I forget to add tape before I die cut, I do the same as Indy_Patti and add Elmers type glue with a toothpick. I find lettering tends to drop off if I use a Quickie glue pen.