If they considered it, was Washington's example cited or did they decide not to run for other reasons? If Washington's tradition was considered so sacred already at that point, what changed that some later presidents were open to the idea?
Great question! I can't answer for Andrew Jackson or James Monroe -- they're a bit out of my area of expertise.
Jefferson, and Madison whom followed him, never appeared, at least publicly, to desire a third term of office. George Washington's decision to decline running for a third term didn't necessarily create an inviolable rule for office holders -- but it at least established an important precedent that future presidents felt inclined to follow -- starting with Jefferson in January of 1808.
Jefferson did write about this, in his own writings. In December 1807, Jefferson declared that he would not seek a third term, like Washington before him. [1] In a letter addressed to the General Assembly of North Carolina on January 10th, 1808, Jefferson publicly declined a third term when he was asked by them about his future in politics.
Similarly with James Madison, a discussion of a third term didn't even really hit the public floor at all. This is likely greatly influenced by the fact that both two term presidents before him had declined a third term, and the fact that Madison was approaching the age of 65 and was preparing for retirement at this point. If Madison ever did consider running for a third term, he did not do so publicly.
[1] Joseph J. Ellis American Sphinx: The Character of Thomas Jefferson. 1998. pp 238