(Nontrivial) Algorithms for finding the third largest element of a set
For $n&gt;5$, this paper proves that the number of comparisons
deterministic algorithms need is ​$n ​ + ​ \big(2 \cdot \lfloor \log_{2}(n)\rfloor \big) ​- ​\big(\text{an element of } \{1,2,3\}\big)$,
where that element depends on $n$'s position relative to the nearby powers of $2$.

Furthermore, for $n&gt;7$, that paper gives a
deterministic algorithm which achieves those values.

By pages 330 and 331, the average number of comparisons needed is ​ $n + \Theta(\ln\ln n)$.

Theorem 5 is that the average number of comparisons used by that
paper's algorithm is at most ​$n + 5 ​\cdot ( 11 + \ln\ln n)$.
I have currently not located anything else regarding non-worst-case
algorithms for your problem, although I've kept finding more as I go along.