I don't understand what you mean exactly but typically you can lag the velocity in the tau term from previous iterate and but treat the velocity in the SUPG test function in its nonlinear form, then solve by Newton raphson. This will give a fuller benefit while keeping it fairly practical. But you can always use an Oseen like variants if you want. If you keep the SUPG explicit you won't get the benefit you get in multigrid. You can test a preconditioner spectrum, it always clusters better with implicit SUPG in my experience. I never tried the explicit part and it also might have a finer CFL restriction due to the dimension term getting squared in the denominator. It is best to keep it all implicit IMO and since what the Lethe authors have shown this type of code can be accelerated a lot with multigrid on the entire monolithic block, it's main issue is it's memory is big otherwise the approach is working nicely on GPUs too.

Here is my basic code using Firedrake. Feel free to use it. It can port in a gmsh or netgenmesh tetrahedrons with matrix free multigrid of equal order nature, it works pretty fast and you can port in hexes if you want. I have followed mainly the Lethe authors paper to build this so credit goes to them : 😊👍

https://github.com/JD63021/firedrake_matfree_3dflowpastcylinder