One additional test I usually do is to check if I can extract information about its generating function given by $$\sum_{x \in \mathcal{C}_n} q^{\operatorname{stat}(x)}$$ for some decomposition $\mathcal{C} = \bigcup_{n} \mathcal{C}_n$ with $\big|\mathcal{C}_n \big| < \infty$ of your combinatoral collection.

In the best case, you find a known equidistributed statistic $\operatorname{stat'}$ to which you can possibly biject.

One additional test I usually do is to check if I can extract information about its generating function given by $$\sum_{x \in \mathcal{C}_n} q^{\operatorname{stat}(x)}$$ for some decomposition $\mathcal{C} = \bigcup_{n} \mathcal{C}_n$ with $\big|\mathcal{C}_n \big| < \infty$ of your combinatoral collection.

In the best case, you find a known equidistributed statistic $\operatorname{stat'}$ to which you can possibly biject.

I guess an archetypical example would be to have the number of inversions of a permutation, so the decomposition is $Perm = \bigcup_n Perm_n$, the statistic for $n=3$ is $$ inv(123) = 0, inv(132) = 1, inv(213) = 1, inv(231) = 2, inv(312) = 2, inv(321 = 3 $$ and its generating function $f_3 = 1 + 2q + 2q^2 + q^3$. You then might realize that this is also the generating function of the major index on permutations.

Finally, let me remark: FindStat shows you the generating function (such as in http://www.findstat.org/St000018), and you can then automatically search the OEIS for the coefficients of the statistic distribution.