June's second, currently, because in July 3 presidents died on the same day and there's only 2 death years for those 3 presidential deaths. July 4th is not just celebrated as independence from Great Britain in 1776 but fifty year's after the passing of 2 famous founders. The historical coincident which did not go unnoticed by contemporaries of Thomas Jefferson and John Adams only enhanced the young Union of States. Adams 2 worried with his last words that Jefferson lived that 50th anniversary of independence when in the non-telegraph (most recent form of instant communication) era and information moved slower Jefferson passed away 5 hours before.
Sharing that historical day but in 1831 is the last president to be considered a founding father, James Monroe. Then there's Zachary Taylor's death on July 9, 1850, leaving Millard Fillmore to decide upon the Compromise of 1850. July 23, 1885 is when Ulysses S. Grant passed away after finishing an epic memoir like no other. The next day, the 24th, but in 1862, is when Martin Van Buren passed. He, like Former President Grant, tried for a non-consecutive term. Where Grant never got the nomination, Democratic Party founder Van Buren was nominated by the Free Soil Party, a single-issue party whose members would later find a home in the Republican Party founded in 1854. Andrew Johnson died on the 31st months after being sworn-in as a U.S. Senator from Tennessee. Johnson-17, so far, is the only former president to be elected to the Senate and for Johnson-17, a body that 7 years prior had almost removed him from the not-yet-built-Oval-Office.
That's 7 presidential deaths and just 5 days.