Atlantic City has an outside shot of getting to the top spot (8.45") and with just 0.05" more can get into the number two spot in terms of June rainfall. The other locations in the region likely don't see their records knocked out. Reading's record is 15.73" of rain -- set back in 1972 (thank you Agnes).
Around the region, Wilmington's June is the fourth rainiest month regardless on record, behind August 1911, August 2011, and July 1919.
Wilmington's monthly record for this June broke the old record by nearly four inches (9.90", set back in 2003).