What did the Republicans win in 1866 that they were able to over-ride presidential vetoes on many bills?

They had a super-majority (a majority of at least two thirds) in both houses of Congress after the 1864 elections.

They had a super-super-majority (a majority of at least three fourths) in both houses beginning May 8, 1867, but a super-majority in both houses is sufficient to override a presidential veto.