It really depends on the destination. H class tickets are pretty high fare for some destination. The HKG H class tickets that I usually get allows me to make changes to the domestic segment at anytime without change fee and the HKG ticket office always waive the change fee as well for the international segment. I believe you paid close to $3000 for that H ticket? That's pretty close to the next available Y fare ($3700?) so it makes sense that there are no change fee on it. On the other hand, there are some pretty low H fare on YYZ-LHR and they charge $200 to make changes to it. So it all depends on your destination.
Are you still going to upgrade it to a Y fare to upgrade?