Something like this might work, though greyelf's idea is probably simpler/more foolproof.
window.onbeforeunload = function () {
window.sessionStorage.setItem('twine-reload-flag', 'true');
};
postdisplay['refresh-taker'] = function (t) {
delete postdisplay[t];
var refresh = sessionStorage.getItem('twine-reload-flag');
if (refresh === 'true' &&
passage() !== 'Start') { // your starting passage here
Engine.play('somePassage'); // passage to forward player to
}
};
This script might have some blind spots I'm not thinking of; all it does is set a bit of data in session storage when the page is unloaded. It will ignore UI restarts and saving/loading, and because it's in session storage, the data won't persist or anything.
If you use this system, test it out in a couple of browsers; I only tested it in Chrome.