Is there not a vapor barrier under your presumably poured basement? Are there apparent joints/cracks in your basement floor? Are your floor drains, water & power subsurface entry a potential conduit/improperly sealed? Might radon be diffusing from your water (if you have a well, a possibility)? A properly constructed new basement should not let radon permeate randomly through the floor. My suggestion is to see if you can ID the entry, in your scenario it is probably a defined point or points that may be easily remedied. BTW , I'm a few miles South of town too.
ETA: There is much conjecture with the 4 pCi/L air standard. I work with risk assessors who feel it's ultra conservative and lends itself to snake oil biz. You are likey being poisoned more from your carpet, garage, Chinese drywall, all the many volatile compounds brought into the house then any that little bit of radon could do. Generally speaking the air quality in any building is inferior to outside and could be mitigated.