Mathematical Art Project: Tom Hull's Five Intersecting Tetrahedra

[Kernel Sohcahtoa]

Ooh, I will definitely try to do this!
I'll just try to make one out of just blank paper until I can get hold of some colored paper.

That is impressive.  After building your initial tetrahedral frame, I suppose you could assign an imaginary color to each new frame (or make helpful markers on your model), and along with following the tips mentioned in the first assembly strategy of the op (a possible approach), you may indeed have a workable approach (especially if you minimize the shifting of the initial frames).  In all honesty, If I only had blank paper available to me, then I'd break down the paper into the smaller pieces (5 groups, each group containing 6 pieces of paper, 30 pieces total) and then color them in accordingly.  I would then declare the Fit as a training/learning model.  But that's just me.  Good luck Handmaid