John Tyler decided to take the oath of office as president immediately. Already, some people were calling him "vice president, acting as president" instead of "president."
He made it clear that he was fully the president and established a precedent [an example for others to follow] for orderly presidential succession by taking the oath and moving into the White House three days later. No one challenged his actions then, nor has anyone challenged the vice presidents who assumed power under similar circumstances since.
The Constitution was modified, but not until 1967, with the 25th Amendment. It clearly define guidelines for the vice president to succeed the president in a variety of circumstances. Vice presidents have followed the "Tyler precedent" after the deaths of Presidents Taylor, Lincoln, Garfield, McKinley, Harding, Franklin Delano Roosevelt, and Kennedy.