Hence the setup where you have nested mu-metal cans...
Degauss the outer one in-situ, and the fields inside the final shield
are measured in terms of one or two microguass.
In principle, you're right, but in practice some magnetic flux lines get pinned to sites in the mu metal making it very difficult to get below the milligauss level. Degaussing the outermost can is done with an AC field that decreases in amplitude with time, "shaking" the flux lines loose from pinning centers and allowing them to enter the mu metal (i.e. taking them out of region in the middle of the can where you want the low field). However, because the outer can shields the inner cans, the degaussing coil doesn't work on them, leaving some of the flux lines trapped, which is why there is a limit to how much shielding is possible no matter how many cans are used. Before someone suggests it, no, degaussing the inner can, then adding a second can and degaussing it, then adding another can, etc. doesn't solve the problem, because when the subsequent cans are added they carry with them flux lines, some of which are not perfectly shielded by the innermost can and that get trapped by that can.
To do better requires can in the center made of a superconductor. First the mu metal reduces the field as much as it can, then the inner shield is cooled below the superconducting transition temperature, at which point "all" remaining flux lines are expelled ("the Meissner Effect"). Once again, though, pinning rears its ugly head, and even superconductivity doesn't reduce the field to truly zero in the center. Some superconductors are better than others at having minimal (but, unfortunately, still not zero) pinning. None of which helps the OP find the 0.05" mu metal he's looking for...