Second $X$ problem is the problem of deciding the existence of another solution different from some given solution for problem instance. 

For some $NP$-complete problems, the second solution version is $NP$-complete (deciding the existence of another solution for the partial Latin square completion problem) while for others it is either trivial (Second NAE SAT) or it can not be $NP$-complete (Second Hamiltonian cycle in cubic graphs) under widely believed complexity conjecture. I am interested in the opposite direction:

> Does “Second $X$ is NP-complete” imply “$X$ is NP-complete”?