[HexWiki](https://www.hexwiki.net/index.php/Basic_(strategy_guide)#Board_size) says a typical (human) game fills roughly 1/3 of the board, but people probably resign when a virtual connection is formed or earlier.

If you want to do some empirical study, there is [a version of KataGo](https://www.hexwiki.net/index.php/KataHex) modified for Hex by HZY, and KataGo allows training networks that work for multiple board sizes, but I don't know if the code uses the number of moves as an auxiliary target in addition to winning. Networks trained on up to 27x27 board are available [here](https://drive.google.com/file/d/1VfWL1MUOCSQo78KdgEkF7roErRl-Wjhp/view) (together with GUI) and the code is [here](https://github.com/hzyhhzy/KataGo/tree/Hex2022), but apparently HZY has started working on [an update](https://github.com/hzyhhzy/KataGo/tree/Hex2024).

Even though all 9x9 openings and two [10x10](https://webdocs.cs.ualberta.ca/~hayward/talks/hex.sol10.pdf) [openings](https://webdocs.cs.ualberta.ca/~hayward/396/hoven/8hex.pdf) are solved (from 2013; any updates?), the solutions probably optimize the number of steps to a virtual connection (e.g. consisting of templates) rather than to an actual connection; but maybe this is close enough.

(Curiously I arrived at this question by searching "percolation theory" and "Hex". I saw HZY's empirical study of the winrates of openings [on Zhihu](https://zhuanlan.zhihu.com/p/476464087), and wonder whether there's a percolation theoretic explanation of the observed pattern, and whether one can exhibit a winning first move on every (large enough?) board and prove it's winning; because there's always a winner, one only needs to show it's "at least as good" as all other moves.)