How does “out” function in “next time out”?
Solution 1:
Without reference or other support for the whole phrase, I think it may be derived from out of the gate, a horse or doge racing reference.
There are many analogies in American politics to horse racing (nothwithstanding its charactierization as the "sport of kings"). This seems to be one of these. The full phrase would seem to mean
Do you think she will run for president next time out [of the gate, that is, in the next race]?