How do they avoid ending it?
It was already ended in all the states except LA and SC even before the Hayes inauguration. The House of Representatives was controlled by the Democratic Party, who of course would not vote funding for military action in the South. By 1876 the US Army was down to 27,000 men, most of whom were needed out west. Only about 3,000 were left in the South - too few to make much difference.
As the First Lady was to say to a critic of her husband's policy "What was Mr Hayes to do? He had no army".