I've looked at some tutorials and I'm now using Blender's physics simulation to create better tape. It's much more realistic now and much closer to how I want it to look.
Later on I'd like to add the option of adding a piece of tape to other gags (like balls), although I'm not sure what effect that will have on gameplay aside from looking awesome.
I made the gag silver so the texture shows up better. The finished gag will likely be black by default. In the future I'm going to add the ability to pick a custom colour.
This is how the gag currently looks in Blender. Unity might not support all the material features that it currently uses, but it should look pretty similar when I export it.