The subject of maximal inequalities exactly concerns bounds that improve upon the union bound. These started with Hardy-Littlewood in analysis. Perhaps the earliest example in probability theory is Kolmogorov's inequality [1] (which improves on Chebyshev's inequality followed by a union bound. )
Later came Doob's martingale maximal inequalities (see e.g. [2], [3]).
Another example is the Maximal_ergodic_theorem [4] which is a different improvement on Markov inequality followed by a union bound. One of my favorites is Starr's inequality [5], see the appendix of [6] for a short proof. A novel direction in discrete spaces is [7].

[1]https://en.wikipedia.org/wiki/Kolmogorov%27s_inequality#:~:text=In%20probability%20theory%2C%20Kolmogorov's%20inequality,the%20Russian%20mathematician%20Andrey%20Kolmogorov.

[2] https://en.wikipedia.org/wiki/Doob%27s_martingale_inequality
 
[3] http://bass.math.uconn.edu/math5160f14/post4b.pdf

[4] https://en.wikipedia.org/wiki/Maximal_ergodic_theorem

[5] Norton Starr. Operator limit theorems. Transactions of the American Mathematical
Society, pages 90–115, 1966.

[6] Basu, Riddhipratim, Jonathan Hermon, and Yuval Peres. "Characterization of cutoff for reversible Markov chains." The Annals of Probability 45, no. 3 (2017): 1448-1487. https://arxiv.org/pdf/1409.3250.pdf

[7] Harrow, A.W., Kolla, A. and Schulman, L.J., 2014. Dimension-Free L2 Maximal Inequality for Spherical Means in the Hypercube. Theory of Computing, 10(3), pp.55-75.