Measure the air-pressure dependence of muon flux -- this requires a very large number of counts per minute.
Perform a higher-resolution passive muon radiography experiment to map out the mass density of Tate Labs
Attempt a shielding-free measurement of the angular distribution, e.g. in a wooden building, top-floor room, or open field. In this case, it is necessary to use a small amount of lead to shield out electrons.
Measure the dependence of the angular distribution or even energy spectrum of cosmic-ray muons on the amount of shielding; this would require setting up the apparatus in a sequence of laboratories on different floors or setting up the equipment outside and using varying amounts of lead shielding.