HexWiki 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 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 (together with GUI) and the code is here, but apparently HZY has started working on an update.
Even though all 9x9 openings and two 10x10 openings 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, 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. Let me mention that there's a explicit (polynomial time) winning strategy that starts with two stones on boards up to 10x10, three stones on boards up to 16x16, and so on. Moore and Shannon's resistor network is used in programs like Hexy as an evaluation function, and apparently there are papers that study resistor networks from a percolation theory perspective.)