Ok, I found a trick with the "Next" button.
In the first step, after selecting the service, with or without provider change, when you click Next, the program updates the available hours again.
I added theses lines in /assets/js/frontend_book.js in the "Next Step Button "Clicked" section :
$('.button-next').click(function() {
//If we are on the 1st tab then update today's available hours
if ($(this).attr('data-step_index') === '1') {
FrontendBook.getAvailableHours(Date.today().toString('dd-MM-yyyy'));
}
In case it could help someone.