Like the city’s summer temperatures, Chicago’s suburban housing market continued to heat up in June with a rising median sales price and fewer days on the market overall, according to The Mainstreet Organization of Realtors’ (MORe) report on the June market. MORe’s report on the May market similarly described substantial increases for real estate in 200 communities across DuPage, Lake and suburban Cook counties, but its recently released data showed these markets getting even stronger in June. The median sale price across the studied Chicago suburbs moved up 7.2 percent year-over-year to reach $268,000 in June. In May, the median price was $249,900. Not only are homes selling for more, MORe says they are moving off the market at a quicker pace. Time on the market dropped 6.7…