The question of whether a Sasakian metric can admit an additional compatible ($K$)contact structure is addressed. In the complete case if the second structure is also assumed Sasakian, works of Tachibana-Yu and Tanno show that the manifold must be 3-Sasakian or an odd dimensional sphere with constant curvature. Some extensions of this result are obtained, mainly in dimensions 3 and 5.
"Sasakian Metrics with an Additional Contact Structure." Afr. Diaspora J. Math. (N.S.) 14 (2) 118 - 133, 2012.