Finding yourself needing to cancel your hotel room but you are too close to the arrival date to not cancel without losing money? There is a two step process that you can try: Call the hotel front desk (not the toll free line) and let them know that you need to change the date of your reservation.
Make sure that you call during regular business hours (9-5) as the night shift will usually tell you to call back in the morning.
After explaining what you need changed and if they say that you can't, try the to make the situation more human; ask for their help and give a reason that they can relate to ("Could you please help me? I know that this is close to cancelation but my mother is really sick and I'm stuck. Is there anything you can do?"). Make sure that the fact you want to change the date, not cancel is clear.
If they still refuse to move the date then you might be speaking with someone who is by the book, new, or is on their last leg with the company so they don't want to give them a reason to get fired.
At this point, it would be time to bring it up to management. There is no point in trying to fight or yell as this will get you nowhere. Some people will try to bribe (I'll give you a $20 when I get there) but front desk staff want the tip in hand, not the promise of one. Give it one last try, sound defeated but friendly ("There's nothing you can do? Thank you for trying. Who would I need to speak with to escalate this? Are they available? Could you get them? Thank you.") It's important that you are not angry as the angrier you are the less likely they are to help you.
Also ask for the supervisor / manager's name as well as extension in case you get disconnected (or you get their voicemail).
When you do speak to the supervisor / manager, make sure to use the same reason you did with the front desk and stress how stuck you are as well as how much you need their help to change the date on the reservation. Asking for help does work wonders compared to demanding something.
If they cannot change the reservation then the property either has had too many cancelations that month or the date you are wanting is sold out. Ask them what they can offer you; a different date? discount? free cancellation?
If they can change the date (make sure you put it out a few weeks to a month later than the actual reservation), double check the cost as well as cancelation fees and room type to make sure the reservation is "correct". If you just ask about the cancelation policy, that is just a big give away.
Once everything has changed, wait a couple of days and then call the toll free line to cancel. Don't call during regular business hours as the number might go directly to the front desk and you don't want to risk speaking to the people who you just asked their help for to make the change as they might not allow you to cancel. However, you won't be within the cancelation period so it should be done easily unless you agreed to a non-refundable / non-changeable rate.
Also, a hotel front desk can make notes on your reservation as well as your master profile (yes, these are made when you create a reservation and added on to with every booking unless you change your address) so if you have a really good note taker or annoy the person you are speaking with then everyone will have all the details regarding your conversation.