Seems like the ideal school for Rhodes college competition would be one that:
(1) is located in a relatively unpopulated state, with little or competition from other elite schools;
(2) draw lots of students from highly competitive states (e.g. CA, MA, NY), who could compete in that relatively unpopulated state, instead of their home states.
This basically describes Reed College. It looks like the only other school in the state that has produced significant numbers of Rhodes Scholars (more than 3) is University of Oregon. And Reed is noted for drawing “long distance” students; they were recently ranked #1 in the “students farthest from home” category
http://www.rhodesscholar.org/assets/uploads/RS_Number%20of%20Winners%20by%20Institution_1_30_17.pdf